Answer: because sometimes the words come first, to point your heart in the right direction.
Answer: because it's the right thing to do. We try to do the right thing, whether we feel like it or not.
Lately I've re-introduced the concept that sometimes it's politic to apologize more than once. Anger dampens one's ability to hear, you know. It sometimes takes more than one I'm sorry to get a hurt or angry person's ears to process what you've said. And repeating I'm sorry leads to peace much faster than repeating the sacred litany of What He Did First. (Silly Mom! Who's seeking peace?)
If we ever master the basics, I've got a harder idea waiting in the wings: You can, and should, apologize for your share of the problem without regard to what the other person has done, and whether or not the other person apologizes in return.
How long do you think it will take to get there? A long time. Last week I was forcefully reminded that there are a lot of adults who still haven't arrived. (I'll spare you the details.)
It got me to pondering. What do people think they have to lose by saying I'm sorry? An apology doesn't affect your bank account. It doesn't cause cancer. It's healthy and mind-clearing and helps everyone involved.
When you have the wherewithal to omit references to how the other person was (also) at fault, you are heard. You are heard in a way that tends to calm the waters quickly, and sometimes that calms the other person enough to hear her own conscience. Not always. But often. Which is a good thing.
But for now we're still working on saying I'm sorry without snarling. Practice makes progress.