As a teacher and parent, the child who hits, bites etc was calmly and firmly told, "People are not for hitting/biting etc," and encouraged to think of a way to make amends after a sincere apology. They also learned that their "Sorry!" was not a re-sent button but carried the commitment not go repeat the behaviour.
Often it takes a little time for the hitter to be ready to apologise but they spend that time alone calming down and thinking, not getting adult attention, which is focussed on seeing the victim is OK.
Children do need to learn how to apologise and put things right; it isn't innate or developmental and some don't witness it from their families.
Modelling how to apologise when in the wrong is an important part of teaching/parenting!