It depends on the context. As Paul explained above, "You got me," could mean "I don't know."
You may also hear "You got me" in American movies when someone is finally caught by the police or someone chasing them. For example, if someone keeps robbing banks and a police officer finally arrests that person, the robber may say to the officer, "you got me" (to mean "you finally caught me"). You may also hear a movie character say (with a great sigh before dying) "you got me" after he's been injured by the person chasing him.
In the US, people using slang may say, "I got you" to imply that they are there to support a friend, or to imply that they understand what someone is saying. Both of these uses are very informal (and are regional slang), so I wouldn't use them in a professional setting.
Friend 1: Hey, will you help me move out of my apartment this weekend?
Friend 2: I got you. (When spoken, people pronounce it as one word, "Igotchu.")