What can I say to people when they ask what I did to my wrist when they see the cuts I put on it?

What did you do? If you tried to hurt yourself, or if you did cut yourself it would be wise to discuss this with a doctor.
Tough answer. It is really nobody's business, but people are nosey. As a plastic surgeon, my staff and i frequently help post-op patients come up with answers for inquisitive people. If you replied that you were in an accident and sustained multiple cuts, it may get them off your back. More important, though, is that if self-mutilation is still a problem, you need to get professional help. Best of luck.