Paybacks are scored when you crash someone who has crashed you. You'll know they are eligible for a payback when they have a green square above their car. You may unlock this naturally throughout the Championship Mode, if you pay attention to the green square cars and ram them with your boost. If you don't have 10 paybacks by the end of Championship Mode, the 2nd tier (Pro Event) of races has a Bonus Event (Payback Challenge) that is specifically designed to get a payback easily. Just grind out the rest of the paybacks you need here.