Sometimes though you didn't actually do something wrong and this is a way to acknowledge that you hear another person's upset but you're not going to admit fault.

For example (this happened to me):

My mother: When you support gay marriage it makes me so upset, I want to die!

Me: I'm sorry you feel that way.

In this case I'm not sorry for doing it but I'm sad that she's sad. Doesn't make it my responsibility though.