What it means to me is that the goal of aikido is not victory over another, but victory over your own selfish desires/wants. In other words, the true opponent is within, not outside of us. In fighting and overcoming our own selfishnesses and weaknesses we can more fully appreciate those of others and the struggles they face. That can make us more understanding, rendering us more tolerant, and less likely to take part or initiate situations requiring aikido. And, in the event it's needed to do so in a more caring way, rather than seeeking dominance in an ego sense. Make sense?
P.S. On the left shoulder of my gi I have the kanji for "masakatsu agatsu" to remind of that each time I put it on. On the right shoulder I've just added the kanji for "hibi shoshinsha" (not sure if I'm remembering right about the "sha") which means "every day a new mind" or "beginner's mind" which I'd like to think of before I get on the mat.