'You got me' Can mean a number of things. It can be as you are playing a game, whether physical or a board game, and your opponent gets a score, going against something you did perhaps. It can also mean you do not know when a person asks you a question.

If it's a question, then it may mean: "Did you understand it?" (=Did you get it?) or if it's a statement, then "You understood what I meant to say" (=It's right)

Example: - Do you know the topic of the paper on which Einstein got his only Nobel Price? - You got me (meaning: your question has puzzled me). What is it, then? Man, "you got me" can be a really confusing statement sometimes.

It can also mean that someone discovered a secret that you were trying to keep, or discovered that you were trying to fool them. Equivalent to "okay, okay, you caught me". "Got" is a terribly non-specific word.

It can also mean you've won or beaten me [ in a game, sport]