What does it mean to humble yourself?

For the past few weeks, I've pondered this question way too many times. It had been weighing heavily on my heart and I didn't like one bit. I tried to surpress it as much as I could, but it just wouldnt go away!

You see, in the past few years, there has been a lot of conflict and discord in certain areas of my life. I didn't want to "talk it out" with a therapist. I tried to resolve it on my own; but to no avail. I just couldnt find any sort of resolution that would give me the peace I so desperately wanted. So I sought out God in prayer and asked for guidance.

No sooner than asking, I stumbled across this passage:
"Think of the person who causes the most conflict in your life. Would you like to begin to resolve the conflict with that person? Are you willing to apologize for your part? Maybe he or she is 95 percent at fault, but your responsibility is to take care of your 5 percent and let God handle the rest."

This really hit home for me. You see, I thought humbling myself to those I have hurt would mean I was somehow giving up power. Giving them license to get off the hook for their part.
Admitting to myself that I was wrong was hard, but admitting I was wrong to the offended party was even harded. It was downright terrifying! Who ever said the first step is the hardest, wasnt kidding! But it's also said that God gives grace to those who humble themselves.
I hope I can find grace and forgiveness, and not become a doormat, as I journey through this uncharted territory of humility.