D&D is an RPG. A role playing characters play a role in a story. The story, and game, is at its best when the story is immersive.
A world that makes sense: There should be nothing that throws PCs out of the moment because it does not make sense to the players. Monsters and NPCs need food sources, refuse options, water options, and a reason for living. They need to adapt to their circumstances. If the monsters in Room A, which is on the other side of a wall from Room B, get in a fight, the monsters in Room B should hear it and respond to it.
A good story: The story also needs to be a good story, and that means that the protagonists need to be involved in it. Their backstory, their personal motivations, and their goals should all be tools you use when crafting the campaign. If they escaped from a slaver, that slaver, like Chekov's Gun, should be used in the story.
What if the PC dies? Doesn't that leave that story element dangling? Maybe. Or maybe their ally knows their story and avenges the prior misdeed.
What do you do about the background of a new character introduced mid-campaign? Do you try to work in their elements? Sure! You might even play a heavier hand in helping them craft their history once the game has started. If a player intrduces a new PC once a campaig has begun, we roleplay their back story a bit - giving them a chance to experience that story and for me to introduce story elements that already fit into the game.
Your players are going to have more fun in a world where their PCs matter.
As an example: I ran a short campaign for four players last year that began in a smallish city. The first PC was a rogue that had been thrown out of the 'thieves guild' (all three members) and was told he had to leave town. The second PC was the local cleric's son, but his father had not returned from his last trip North to the big city. The third PC, a farmer that used magic to farm, had sent his son to help the Cleric on the jourey from which he had not returned. The final PC was a druid that lived outside the village in the woods that had found a relic in the woods and needed to research it in the big city.
They all had a reason to head North to the big city. Along the way, they found the camp of the cleric and son, abandoned, and determined that they'd been set upon by a pair of brigands. Once captured, they'd been taken North to the city. Later on they'd discover those two brigands were the former guild mates of the rogue, that the Cleric being kidnapped worked into the story of the relic, and that a lot of their other background, flaw, etc... elements were meaningful. They did not all unfold in the first few adventures, but there was a place for all of it and it made the game better. It certainly added to immersion that the PCs had elements from their past that 'fit' in the rest of the world.
And this is how I've run games for a long time - I mine the backstory. There is a reason why they added backgrounds into 5E to provide easier ways to do this....
