I think the key here is that people function better, and break addictions better, when they realize and submit to the concept that they do not (and can't) have complete power and control over their own lives. The notion of a "higher power" or "God" can help people get into that posture of "letting go", which helps them find real support outside of themselves (and their addictions).

Being a Christian myself, I know I function much better when I'm consciously in an attitude of trusting God -- it lets me relax and simply be who I am and do what I need to do, without worrying about "fixing things" outside of my control. (And yes, I do take an intellectually rigorous approach to my faith, but that's a different topic.)