What should you do if you saw a big piece of fur hanging from your cat's testicles when he was outside the night before?

That depends, is it his own fur? Cats, particularly male cats, are very protective of their territory and they'll fight to gain MORE territory. Most cat fights, between males especially, is about having the breeding rights for a specific territory and if you bring your cat into some other cat's territory, there will be a fight. Females will fight over "feeding" territory, but they aren't generally as viscious. If your cat has sores or starts to develop infected sites where it has been bitten or scratched, you'll need to take him to his vet. Otherwise, leave him alone. The catfights are their way of saying, "This land is my land."