A frequent occurrence in many campaigns is tying the plot and characters into the players' backstories. I recently decided to do this and am having a bit of trouble coming up with how to structure it. I want the story to be meaningful, but I don't want the other players to feel like one of them has taken the spotlight because it's their lore. How do I balance doing this?

It's a tricky balancing act. The way I did it was to have a central narrative that wasn't related to any of their backstories, but have that narrative interact with their backstories for the PCs that wanted that. This ensured that while there were "chapters" where one PC was more of a focus than the others as they were travelling in their homeland and dealing with issues affecting their backstory, but that none of them were ever the primary protagonist, and each of them got a turn. 

I agree with the person above. Having an overarching story helps keep everyone invested. Having your players arc intertwine with the story, or be parallel with it, like having the destinations be the same, can make it far easier. You could also make an entire arc about a player. It's always a good idea to tie the other players motives to it, like the PC arc is to overthrow a dictator that so happens to be their parent, and the other pcs think "Yeah, screw that guy." If you're worried about spotlighting though, wait for the dedicated arcs until the party is closer. A party is more willing to do a PC arc if they like said PC.

I should also add, as a player, I absolutely love experiencing character arcs, even if they aren't mine. If you have players there to play, they will more than likely be happy to participate. If you balance it a little and give everyone a turn, they will very likely be happy. Especially if you work with the players to get them excited to play their arcs. Communication is very valuable. If you're worried, talk with players, or get them excited. Both are very effective tools

That's how I run my campaigns. I build a world then weave their stories into it. There is no world ending event or bbeg out to destroy everything. The players all know about it as well. So they all understand that we are focusing on a players story. They also know that their backstory will be coming up as well. 

The players who have had their backstory completed are not forgotten. I always add in world events that is a result of their story. Like the changes in the world and how what they did affects the world. That way they see those changes and don't feel forgotten because their story is complete. 

 Internet is trash out here ATM.

If you blindly ask the player for random backstories then you're going to get a mixed bag. If you at least give them some world information and a starting point then the backstories can meld together much more easily.

I enjoy playing character arcs for other players - doesn't feel like I lose anything from that.

One thing to watch out for, and maybe this is obvious, is that you'll need that player to be present for every session of that arc or it won't work.