How should I repair a burned outlet?
First, let's address the cause of the problem for safety. Something was drawing too much power for the wiring, but not enough to trip the breaker. This means that there was a weak link somewhere. I suspect either a bad cord end or a faulty outlet. If the outlet was old, it probably had a weak grasp on the cord prongs. 

The outlet must be replaced. It has been damaged and is a further safety risk. It's an inexpensive fix, but one that requires a basic understanding of house wiring and connection techniques. Many tutorials can be had online, but the details of your repair depend on some things you haven't shared, such as the age of the home and the outlet type.