I think the quote means that an individual who achieves success through his own abilities, talents, and hard work (own wings) has no limit as to where he/she can go; kind of like the saying "the sky is the limit."
I think it also might suggest that this person is free to feel proud of their own success because it is their own. They didn't need to have others carry them on their shoulders, so they can take pride and all the credit for the work that is their own.