It means: "Don't annoy me." "Don't make me angry." It's something you'd say to someone who was bugging you, or teasing you, or otherwise getting on your nerves---warning them that you might lose your temper and respond in a way they won't like.
-Hey, Why are you drinking so much, dude?
-Don't mess with me.
Friend 1: Bro this place is kinda dangerous, what if we get robbed?
Friend 2: Don't sweat it bro, every city is dangerous when I'm in it, I keep that Nina with me like a Guard!