How does one get over their shame from the hurt they cause someone else?

Shame. Genuine repentance and expressed apology and of course not repeating the hurtful behavior.
Time & action. Shame, like pain, is one way we know something's off about our body or mind. Both can be quite helpful signals but also can become problems themselves. First step is to identify the source and decide if there is anything you can do about it--now or for furture similar cases. If you're stuck, a brief case of therapy may help, especially if this feels like a pattern for you.
Shame. Shame is just about the darkest emotion people can face. It presupposes that there's an audience to our alleged wrongdoings, and that devastating judgement will be passed. Something to remember is that everyone has something they feel ashamed about; it's inescapable. Find a reputable psychotherapist to speak with. Getting it off one's chest in a safe, nonjudgmental place is the first step.
Shame. I think maybe i answered this question earlier. A different take on it is that if you have access to the person you might discuss it with them ; apologize. You may have hurt them because you were hurt that is what shame does to us. If you have access to a therapist or even a good friend to talk to that would help. Guilt is a consequence of shame and i don't think it does much good.