When you're deciding who will pay for a meal, the phrase "I've got it" means "I'll pay for the meal":
A: How much is it?
B: Don't worry about it. I've got it.
In a more general sense, you say "I've got it" when someone is offering to help with something, but you want to show that you can handle the situation without any help. For example, if you're carrying a heavy-looking suitcase and someone offers to carry it for you, you say:
That's all right, I've got it.
You might also hear:
I got it.
This isn't technically right, but it's common in American English.