Hearing, "I'm sorry" can change the way that we look at a person who has wronged us. Ninety percent of the work of apologizing is done before we even begin talking.
An effective apology is less about saying that you are sorry and more about feeling how the other person feels. You understand what you have done when you can name the cause and effect. You must find the cause of the damage and step inside that situation from the other person's perspective.
Remorse is the feeling of wishing that you could take back the thing you did. Remorse means that you recognize that you were wrong. Your remorse is about your actions, not about the effect of your actions. Dive deeper into the situation; ask questions to discover what that person feels the way they feel.
Let your remorse fuel the delivery of those three important words, "I am sorry."
Apologies require humility. You cannot be both right and wrong at the same time. You can apologize for not responding to the way a person was acting. Apologies require a lot of empathy. You've received good apologies before, so keep those in mind.