I can't remember where, but I was just doing some reading on this. I think the idea is that the intense pain of the injection causes your body to release more endorphins - the body's natural pain relief hormones, which makes the contractions easier to bear. From what I read, I gathered that the exact science of how it works hasn't really been figured out yet.
I don't know what your labour is like, but from the sounds of it, I think I'd pass. By the time the contractions get to the really really hard "give me drugs RIGHT NOW and GET THIS BABY OUT!!!" stage, you're almost done, so why bother adding to the pain? But, hey, everyone's circumstance is different, right?
Curious to hear from some who've tried it.