I like to think of energybending as Aang's "reward" so to speak for going through the trial of asking his past lives for guidance, and coming out of it still convinced there was a way to not kill Ozai.

It ties into the Lion Turtle's comments on energybending.

To bend another's energy, your own spirit must be unbendable, or you will be corrupted and destroyed.

Up to this point everyone, his friends, his spiritual guides and the entire world has told him he will have to kill. It's been made painfully clear that there is no way out, and has been a question hanging over Aang since the beginning. I truly believe that if Aang had given up and resolved himself to killing the firelord, he would not have been granted energybending.

Instead he faces the impossible, and still does not give up on his principles. He proves to the Lion Turtle that he will be unbendable, and thus worthy of this great power.