Engaging PC Backstories
Backstories are the R in RPG: they help the player define what role they will play in the game by describing what sort of character they are. They usually have a story, but really, all the story, all the characters in their life, all the experiences, add up to what should be a guide for how they will play. But we want to know how to use backstories as well.
So I'll do a little prep work in defining backstories, recommend how to get ones you can use from your players, and then go over some options for how to engage the player by activating that backstory.
Backstories
To put simply, a backstory is not a gag or gimmick, but an story of what defines the character's motivations to journey and the decision cues they use to resolve conflict.
What Isn't a Backstory
Accents and such are gimmicks; they can be part of a characterization, but they are not a character. That your character speaks like Sean Connery does not describe anything about how they will make their decisions in-game.
Gags are not necessarily outright jokes, but stories that create fluff for the mechanics, like a fighter who uses only pots and pans for armor, or a sentient bag of garbage. Again, this can be part of the characterization, but it is not their story.
But there are still two parts to my definition, and without both, it isn't really a backstory. Let's take the most stereotypical backstory we can find: "My parents/tribe/village was killed by mysterious forces and I need to bring the killers to justice." That's the plot of like, 90% of fantasy protagonists. That is only the motivation to adventure though. It does not help the player understand who this character is and how they make decisions about conflict.
Let's take another stereotypical example from the other direction of just how they make decisions: "My rogue is a kleptomaniac and tries to steal everything!" Now we have a crude explanation of how they make their decisions, in that they are selfish and when presented with conflict will generally act so. But that does not explain why they are out adventuring, just that they're a petty thief.
What Is a Backstory
So a backstory should define their reason for adventuring and provide cues about how they handle conflict and decisions. When you put both together, you very quickly get a fully-fleshed character. Let's go back to our two stereotypical examples and combine them. "My family was killed. I steal and hoard every penny I can get because I am saving up to resurrect them." We have a reason they have joined the party and gone out adventuring, we have cues for how they will act; that's something a player can run with pretty easily.
I'll bet that, in the comments, we could very easily make a 2d20 list, 1d20 adventure motivation, 1d20 decision motivation, and ta-da, 400 characters that all make sense and be eminently playable. I'd argue that just about any two combinations, even if they sound miserable individually, will add up to create something greater than the sum of its parts. Maybe not a character with lots of depth, but good enough to play.
That's what the player needs anyway. How about you, the DM?
What the DM Needs from a Backstory
We always talk about Session Zero, and for good reason. Players can easily go wild with their backstories and create things that are not good for the world, or the story, or just the table, or are just really overly intricate for a low-level adventurer. I don't mean to shame people that really like writing character stories, because our real worry, as DMs, is that they don't give us anything that we can use. We run session zeros because we want to encourage them to give us something we can use. Working with your players to fit their characters into your world is just a helpful bonus.
So when you are working with them during a session zero, you are helping them to build a character that they can play, and that you can use to engage the player in the story. If their backstory does not matter to the story, then what's the point of you even reading it? Because in the end, they aren't doing a backstory just for their fun, we are doing it so that we can pull it back up later to engage the player in whatever is happening in the story.
As a DM, you actually need three things from your players: the previous two that players need in motivation and decision cues, but you will also need to add NPCs. When running your session zero, ensure that you get all three from each player. They can build and add as they go, that's up to you and them, but aim to get them all before too long. Much as I like the Session Zero Guide linked above, I usually only have three prompts:
Why have they started adventuring? (Motivation)
What single trait most defines them? (Decision cues)
Give me three living people that your character would have an emotional response (good or bad) upon meeting. (NPCs)
Even if you're mid-campaign and it's a bit late to just go ask these of your players, it's a useful thought experiment to figure them out based on what you know of them so far. So make that list. And now that you have gathered up a list of backstory elements to help your players play, let's go over how you the DM can engage the player with each character element.
Using Backstories to Engage Players
Engaging Players by Engaging Characters' Motivations
These are both the easiest and the hardest. It's the easiest because it's totally clear what you need to do; let them tell their story. If they're looking for justice for their slain family, well give it to them. It's the hardest too because, for most tables, you have as many separate motivations are there are players, so if you focus on one, you leave the others out. Also, for most tables, you have an idea of the story and the BBEG already and it probably isn't also the same as what they have in their backstories.
Your goal is to, therefore, let each player tell their part of the story as it becomes a part of the larger story between you and the other players. You do that by adding their motivations to the story you are building, and linking it with the other players.
Harder said than done, definitely. Let's use our example again, "something killed my family." They're the BBEG, done. Or, it was done on the BBEG's orders. Or it was the BBEG's people just being corrupt and awful, because the BBEG is corrupt and awful and even if it wasn't their direct fault they enabled a broken system.
Let's mix it up with an "heir to a lost kingdom" cliche. Their kingdom was taken over by the BBEG. The BBEG is their uncle who usurped their father. One of the BBEG's minions destroyed the kingdom.
I'm not mixing it up just to provide new examples of basically the same advice, but because we want to see how player stories mix. The easy solution is that the BBEG made both problems and therefore motivations to adventure. But there's other options where they are totally separate, but advancing in one informs the other. Each are after a separate BBEG minion, and each contains a part of the information necessary to defeat the BBEG. The minions that killed the village each took symbols of office and birthright during their capture of lost prince's kingdom, so defeating them also recovers the things he needs to prove his lineage. In a long campaign, each character should have their chance to be the spotlight for a little while and tell their story. But it should be done in parts, not all at once. And at the end of one part, the story does not point to the next part of that PC's, but to the next PC's story, rolling through them all and advancing the main story as each goes. Hopefully, all stories, PCs' and main, conclude around the same time.
Engaging Players by Engaging Characters' Decision Cues
This is where we get moral confusion and productive party conflict, and probably the most rewarding part of DnD, for me anyway, and I suspect for some of my players. Our goal here is to challenge them to think about their decisions and the way that they make them by making the obvious choice a hard one to make. They have to commit to doing something against character, or further commit to their decision cues even when it feels wrong, either way redefining the character and building the number of decision-making cues they have in their arsenal. This is why I talk about how characters resolve conflict; it's not combat or debate, but how they resolve the conflict within themselves. Engage them by challenging how their character makes decisions.
Back to our examples, the kleptomaniac, the good one that is with the reason for their kleptomania being the desire to resurrect dead family. They hoard and steal because they have something to spend that money on. Give them situations where they could save others with their wealth. They might not be able to afford a True Resurrect like they need now, but they could save their friend now with a plain Resurrect. Maybe not even their friend, would they spare the gold to save a child from being orphaned, as they had been? How about the fate of the world, is it worth trading their amassed wealth to the dragon for a mcguffin that will defeat the BBEG?
More importantly, you can challenge them by making situations where multiple sources of decision cues are put into conflict and they must resolve it by considering multiple angles. What does the party do with a prisoner? One may want justice for their family's deaths at the prisoner's hands, but others might need information for their own quest. One may believe in justice, but the other in expedience, which one prevails? You don't have to make the party fight itself on this, though it can be great when it happens right, use NPCs (covered next) to pose different viewpoints of the same situation. They could be having this argument about the prisoner with the king in charge.
Put them in conflict with their own history of decision-making, offering other viewpoints to consider, and see how they grow, either changing and growing in new directions, or stiffening into an even more resolute form of what they began as.
Engaging Players by Engaging Characters' NPCs
Probably the best overall way to get players involved, and the easiest for me to write a guide about, are the NPCs that PCs give you. This is why I ask for an emotional response, rather than any specific NPCs, because I want to know who gets their passions going, be it someone they love or someone they hate. Moreover, everyone has people they love and hate, even the orphan who never knew their parents. That orphan might have been saved only due to the charity of a friendly shoemaker or cleric or whatever, even if the two never spoke, that person is now someone they remember and would defend. They probably knew other orphans who were bullies and kicked them around when younger. Point being, everyone can give you an NPC or three to use.
How to use people they care about? Threaten them. The war is closing in on where they are, do they continue fighting the war or go to evacuate them? The BBEG knows and kidnaps them to get at the PC, or they get enslaved by BBEG armies, or imprisoned for breaking arbitrary laws enforced by the BBEG. Use them as guides. Old mentor sorts can be great info dumps. They can call the PC and party to places that they need to be when they are a little lost as to what to do in the story. They can help pose different viewpoints to a character who usually decides following a single decision cue, challenging them in a friendly way to rethink their ways. Use them to force the PC to commit. They can sacrifice themselves to save the party. They can die to show the malice and power of the enemy, signaling to the party that this threat is beyond them for now. They can give new quests and information that link the PC's story to the bigger ones going around. These are just a few ideas.
How to use people they hate? Make them enemies. It's the easiest thing in the world for a BBEG to get new lieutenants wherever thematically appropriate. They don't even have to join the BBEG, just take advantage of the chaos to advance their own goals, forcing the PC and party to decide which is worse in that moment. Reconcile them. They might have to work together despite the past rivalry for the greater good, but they'll still try to show that they're better than the PC. They could even be in a position of power that the party has to work with to get anything done, forcing the two to somehow reconcile. Maybe they properly redeem themselves in an old mentor-esque moment where the tables flip entirely on the rivalry. Use them to measure change. As the PC grows, both in power and in character by challenging their decision making, bring old challenges back to make them see how they have changed. That bully that beat them up at level 1 is now barely a challenge at level 10. The rival who the PC hurt by being too selfish once wants revenge, only to see a PC who is now more generous. They can exemplify or take to extremes the very same values that the PCs make their decisions with, forcing them to see the potential consequences of their own beliefs. Again, just a few ideas.
Conclusion / TL:DR
So we know what a backstory is: the combination of a motivation to adventure and the characteristic(s) by which they make their decisions during conflict. We also know to get NPCs from them that explain the history of their motivation and give pre-adventure examples of how they resolved conflicts. We've also given a long list of ways to engage all three of these backstory elements.
No guide can be complete. There's so many potential backstories, that nobody can give advice for every situation, you'll just have to work that out on your own. But hopefully with those few key elements, and some ideas on what to target and how, you can use your player's backstories to engage the character, and the player, in the greater story that everyone at the table is telling.