There are many reasons why you should make a sincere apology when you've hurt someone unnecessarily, or have made a mistake.
First, an apology opens a dialogue between yourself and the other person. Your willingness to admit your mistake can give the other person the opportunity he needs to communicate with you, and start dealing with his feelings.
When you apologize, you also acknowledge that you engaged in unacceptable behavior. This helps you rebuild trust and reestablish your relationship with the other person. It also gives you a chance to discuss what is and isn't acceptable.
What's more, when you admit that the situation was your fault, you restore dignity to the person you hurt. This can begin the healing process, and it can ensure that she doesn't unjustly blame herself for what happened.
Last, a sincere apology shows that you're taking responsibility for your actions. This can strengthen your self-confidence, self-respect, and reputation. You're also likely to feel a sense of relief when you come clean about your actions, and it's one of the best ways to restore your integrity in the eyes of others.