When I was younger I used to make up all sorts of stories that would justify my fear of apologizing. My thinking was twisted; I admit it. Somehow I would always find a way to tell myself the story so that I was right and the other person was wrong, or so I was the victim and the other person the villain. They should apologize because I had done nothing wrong -- or at least nothing as wrong as their offense. That was the story I'd tell myself. It felt important to maintain the supposed upper hand for some reason I neither understand nor remember.
Boy was my thinking ridiculous! Maturity has taught me that it is so much easier to apologize first -- and ideally right away -- than to let the so-called offense linger and the story snowball and take on an (inaccurate) life of its own. Apologizing can create intimacy -- room for all parties to be human and to have that humanity accepted.
They also can deepen relationships. Demonstrating grace brings out the graciousness in others. The big part of them will often respond to the big part of you. And when someone can trust you to take responsibility for yourself (and vice versa), the relationship grows stronger.
Apologizing can even be downright fun, especially when you can laugh at yourself or the ridiculousness of the situation. It removes the pressure that pretending that you're perfect creates. You can screw up, apologize, laugh at yourself, learn the lesson and start over.
You can say even I'm sorry strategically. There are times when I'm not really the least bit sorry about what I did, but I *am* sorry about the misunderstanding that occurred. Apologize for that. It will heal and open doors.
Apologizing first is a sign of power. The person who apologizes sets the tone for the rest of the conversation and even the relationship. I wish that I had understood 20 years ago that in a seemingly weak act there is strength.