Shanli Liu breaks down three parts of a real apology: start with "I'm sorry that I" to own it, end with a period not a comma so you don't slip into excuses, then follow up with a concrete commitment like "Next time I'll" or ask how to do better.
How to apologize. Number 1, begin with, "I'm sorry that I..." Then say what you're sorry for. If you begin with, "I'm sorry that you..." Well, you're doing it wrong. Number 2, end your apology with a period, not a comma. So it's, "I'm sorry that I raised my voice," period. Not, "I'm sorry that I raised my voice, comma, but I wouldn't have done it had you not..." That's not an apology. That's an excuse. And number 3, once you've genuinely apologized, you're going to follow it up with, "Next time." For example, "Next time I'm going to do X, Y, and Z," or "There won't be a next time." Or you can ask them, "How should I handle this next time?" They may not have an answer, but they will appreciate the question. So try that, but follow me.
