This expression can mean:
1) Please don't be strict with me.
For example, if a policeman stops you for speeding, you might say "Give me a break! I'm going to see my mother in the hospital!"
2) or it can mean: I don't believe you. I'm not that stupid.
For example, if the TV news says that the economy is getting better, you might say "Give me a break! All the stores are bankrupt!"
