You use this phrase to point out that you're doing something nice for someone that will have to be "paid back" later.
A: Can you come and pick me up? Please?
B: OK, but you owe me one.
Family members, close friends, or colleagues who are friendly with each other think about favors in this way with each other. You do something nice for your friend and say "You owe me one." Later, when you want to ask your friend to do something that he doesn't want to do, you can remind him: "Remember that time I covered your shift on a Saturday when you wanted to go to the beach?"
The word "one" in "You owe me one" doesn't refer to anything in particular. It doesn't mean "one favor" or "one point" or anything like that. It's just part of the phrase.
You might also hear another version of this phrase, which is even stronger:
You owe me, big time.