I want to preface my answer by saying that I understand where your impulse is coming from, that I respect it at least in part, and that I share that impulse. But with that preface, I must in good conscience push back against the frame of the question:
Consider Reining In Your Urge To Discourage
I respect and share your impulse, here. I really do.
Every time a player of mine goes off and does something like this, I have a little (or not so little) frisson of fear: What if they're going to break something I had planned? What if they're power gaming for an advantage? What if they touch my GM stuff?!
After a long career in GMing, though, it's rarely been an issue in the way I fear. Are power gamers gonna power game? Yes. But it turns out they're fairly easy to detect and shut down. Will the players occasionally do something that wrecks a plan? I can't actually think of a time that's happened, but I can think of several times that a player was pushing in one genre direction and I was pushing in another... which can be an issue.
But on the other hand, having engaged players who want to contribute to the setting is a blessing in many ways. First, it just means that they are active and are engaged which is a great thing in and of itself. Second, it means that in some sense, they're taking a little of the creativity burden off your shoulders. Third, when players do this in good faith, they are telling you what they are interested in! They're directly telling you something other GMs have to read tea leaves and pull teeth to get from their players. And fourth, sometimes (not infrequently!) a creative and engaged player will come up with something really good that you wouldn't have thought of that you can turn to your advantage.
Still, there are at least potential issues.
Meet Them In The Middle, With Approval
What I do, therefore, is to make clear the following things:
- I welcome player input.
- But I need to see it and approve of it before it's canon; that's
part of my job as GM, enforcing the game canon.
- If I veto something, I will whenever possible explain
why, but sometimes in order to avoid spoilers, I won't be able to
explain.
- If this all works the way I expect, then over time you'll have a
better feel for what I'll allow, and I'll have a better feel for
what interests you, and we'll clash less and less. But while I hope
to be more (not less!) easy-going over time, I always retain that
canon-veto.
I've found this works pretty well. I'm only GM of my experience who explicitly spells out the fourth point, but that's certainly how it's worked out when I'm a player in a game that encourages player contributions. It definitely works from my side of the table when I am GMing.
Addressing some issues that have come out in comments, which may or may not make it into an edit of the question: I must stress that all of this regards background information, not questions of current NPC ownership or behavior. Different games address this differently, and even within 5e different GMs will have different levels of tolerance. At the very least, influence on current NPC actions strikes me as a qualitatively different question. (And if you think the player is angling to backdoor that sort of thing in through the submitted background, well... that's what veto power is for.)
Best Answer
I highly recommend the random background generator from Ultimate Campaign. There's several automated tools around to roll it up for you if you don't want to do it yourself. Here is one.
For an example, I just gave it "Non-evil Dwarf Fighter" and got back:
Already you can see a possible backstory from this: 2nd of 5 siblings, born to dwarves who (unusually) took to the sea. Your father was killed in a pirate raid while your mother was close to term with you, and the stress sent her into labor. You were bullied as a child for not having a father, although your mother later remarried (hence other siblings and having both parents). When no one else would step up to protect your ship (from more pirates?) you knew bullies for what they were and stepped forward to protect others in your true father's memory. But at some point, you stole a fancy, expensive necklace from your captain to give to the woman you loved...
Based on how guilty you feel about that theft, you'd determine your alignment, and so on (I didn't cite everything here).
It's by no means a complete story, but it's a skeleton that's very easy to flesh out to a paragraph or two at the least.