The way I see it, he's been cheating on her ("treat you right" = "stop sleeping with other girls"), and she's kept trying to leave, but can't stop giving him second chances ("you never had the nerve" = "you really believe that with one more chance, I'll change"). So he thinks about the situation and realizes that he's been an arse for cheating, and begs her for one more chance - because they both really do like each other too much. And because he's figured out that he really does want to be with her, he'll do whatever he can to convince her that he'll treat her right.
So it's sort of like "god, I'm an arsehole, but I'll change - for you."
The way I see it, he's been cheating on her ("treat you right" = "stop sleeping with other girls"), and she's kept trying to leave, but can't stop giving him second chances ("you never had the nerve" = "you really believe that with one more chance, I'll change"). So he thinks about the situation and realizes that he's been an arse for cheating, and begs her for one more chance - because they both really do like each other too much. And because he's figured out that he really does want to be with her, he'll do whatever he can to convince her that he'll treat her right.
So it's sort of like "god, I'm an arsehole, but I'll change - for you."