But what if that person was offended by something that I totally did not mean? They are talking emotions and intentions into my words! Am i responsible for their feelings?!
When we say, “I am sorry,” we are making a statement about ourselves – I am remorseful, I regret my actions, and I hope not to repeat them.
But an apology is not just about you and your feelings. It is about the person you hurt. You don’t apologize just to absolve yourself from guilt, but more to acknowledge that you are the cause of someone else’s pain, and take responsibility for it.
This means that even if you are completely in the right and really did nothing wrong, even if the other person misinterpreted your words or actions, even if you did nothing to regret, nevertheless if someone else is hurting you need to apologize for that.
Only G-d knows who is right and who is wrong in your case. But we all know who is hurting. So we need to be big enough to apologize.
But beware. If you say, “I am sorry for any pain you felt,” it will come across as empty words, as if he has a problem. You need to arouse true feelings of empathy for them, and real regret for your part in what happened by putting yourself in their shoes. Shanah tovah!
Rabbi Moss Menachem Mendel Bluming and Chabad.org