I approach it in two stages.
First: I look for a common thread in the group's character's backstory. It is great when the players collaborate on their own, but that does not always happen. I do not force it. I want the players to play the character they want to play, not the ones that make sense to me and the current adventure plan. Sometimes there is a good story there, sometimes there is not. I do tap into their backstories for some inspiration for the initial adventures, but the adventure does not have to be tied directly to each backstory. Some times, only one or two backstories may have a connection -- i.e. their uncle sent them, their patron has tasked them, etc.
Backstories provide for me NPCs who act as direct agents for the players, indirect influence in the campaign world, and possible villains to be defeated. Backstories provide campaign details, cities, towns, schools, and churches. Backstories provide cultural events (such as holidays), historical events (such as wars), historical relationships (conflicts between kingdoms, towns, guilds, and schools). These are the player's addition to your campaign world and make the character have more relevance in the game.
The player characters are heroes and should feel important and relevant to your campaign world. As soon as they sense that, they will engaged the campaign world in a more meaningful way.
Second: Let the backstory relevance evolve and progress over time. Some players take time to flesh out their character's back story. They may be inspired by the adventure. Some players may has small elements in their character's backstory that make perfect sense in the campaign as it progresses. Do not be afraid to the rare gems buried deep in a character's backstory and pull that out as a tie in the campaign or adventure. The players will be pleasantly surprised and will eventually develop that aspect of their character's history more.
I think it is critically important to listen to the players as they describe their characters, their actions and their motivations during the game. It tells you much more about their character than what they write down. I have no problem tapping into a stated bit of character history as a seed for future adventures or campaign elements. I found that players who are reticent to write a backstory still have one in their mind that they describe during the game. Use it.