I understand many people view this an in impersonal perspective of human relationships. Let's say I say something that makes you upset. My belief is that it is more rational for me to say something like "I understand what I said upset you and I will work to say things in a different way to avoid upsetting you." This is at least an attempt to prevent this from happening again. I know from experience, that in reality, things go along much better if I say "I'm sorry" and don't worry about the behavior. I can do the behavior again and use again "I'm sorry" and things get better much faster.
Of course, if you are always saying "I'm sorry" that isn't good, though it is a little magic pill that most people are taught to accept to make themselves feel better. On the other side of this, besides no liking to say "I'm sorry", it doesn't mean much to me when somebody says it to me. Why would I care that they are sorry? Does that change what they did? It makes more sense to me that they realize what they did and if necessary, compensate me for what happened, or profess to try to prevent doing it again.
What is "I'm sorry" supposed to accomplish? I don't get it, though I do use it to smooth over relationships because I know that people like to hear those magic words.