Depends on what happened and the person.

Sometimes people want to talk about it, sometimes they want to isolate and recover. Some people don't like to talk about things at all.

I usually wait and give them time to calm down before I approach them. I find a time where they're not busy, say I'm sorry about what happened and ask, would you like to talk?

If they say no, I say OK, and let them know if they change their mind, they can let me know and leave it to them to bring it up otherwise letting it go.

If they do want to talk I listen to what they have to say. While they are speaking I do my best to pay close attention, a lot of the time I want to defend myself or interrupt, but I do my best to only speak when I don't understand something to ask for clarification.

Like cat_banana_hat says I resist the temptation to make excuses. I also don't make promises, I never say that I'll change, or this won't happen again. What I do is apologize for what I genuinely feel sorry. If the person wants me to admit wrong and I don't feel that way, I apologize instead that I'm sorry that when I did x, it made you feel y (usually subbing their words for x and y). I don't tell people things that are untrue to smooth things over. Because I speak from a place of truth, and because I do genuinely care that I hurt them they feel heard.

I also know that if I apologize for something I don't feel is true I will feel resentment, and that will blow up later. It makes me even angrier when they're satisfied and I'm stewing over it. I will risk someone else not being entirely happy with my answer over setting up an environment where I'm upset and likely to lash out again. Basically I go for the lesser of two evils, because generally the other person is more skilled and less volatile than I am. If they're upset they can bring it up later and we discuss it, if I'm upset the roof may blow off the house. The best damage control can be prevention of future behaviors/incidents.

If they want promises, I promise to do my best, and work out with them a compromise I what I know I can do. Although people know me well enough that I do work hard at it, even if I screw up, usually at some point I come through. Usually we revisit each time and maybe redefine goals, it's not always about what I do, but things they're willing to do/tolerate as well.

I don't know if that helps or not.