The speaker is proud that he did not spend his life trying to fit in. He did not say what other people wanted him to say. He did not do what other people expected him to do. He is proud that he made his own decisions. He is proud of his individualism. He "says the things he truly feels" and he never "kneels" to anybody else.
When my wife was a kid, her parents tried to teach her to tie her shoelaces. They taught her a certain way to do it. She couldn't learn to do it that way. She invented her own way to tie her shoelaces. She still does it that way. She ties them with a strange knot that is different from the one everyone else uses. I could say "she does it her own way," or I could simply say "she does it her way."
"I've lived a life that's full"--I've done many things.
"I've traveled each and every highway"--"highway" is figurative, and obviously chosen to rhyme. He is saying he's "gone everywhere" and done everything.
But, he says, not only did he do everything, he was in control of his life and he did everything "his way," no matter what other people thought.