"I've got this covered" (not "I got") means that you have the situation under control.

"I've got this covered."= I know what to do

a boy and a girl are out on a date. When they see an ice cream shop, she says she want to buy some. She orders 2 chocolate ice creams and is about to pay for it. The boy says, don't worry i got it covered. It means that he will take care of the situation and in this case pay for the ice cream. ****what a nice guy

It means you will handle it or do it by yourself. For example, if you were with two friends and one of them asks for help with something, you could say "Don't worry, I got this covered," meaning you will do it by yourself--the other friend doesn't need to do anything.