"you've got the best of me", "you've bested me", or "you got me" all mean the same thing, "you've won/you've fooled me".

You won/did better. You have won. You conquered me in a game/competition/contest.

Depends where you found the example... "To get the better of someone" means to defeat or beat them at something "You've got the best of me" (in a love song for example) may mean that person has access to the best sides of your personality.

I tried my best, but you were better. You won!