Look, I see the argument people have against acheivments and gamerscores (they're pointless bragging rights and don't give you anything). Well there is plenty use for them.
And as an intelligent 360 advocate, I would like to list those reasons for you now.
1. Yeah, they're bragging rights. Everyone likes bragging, and most importantly it adds friendly (usually) competition to your gaming among your friends, which is good.
2. The feeling of accomplishment you get from getting an acheivment, especially one you've been working towards for awhile. Boosts your self-esteem (mine atleast) and adds to the feeling of accomplishment you normally get from completing games.
3. They push you to play more of the game, replay it and do things you normally wouldn't do, letting you get more out of your games and more for your money. Usually I just play through a game once then go on to multiplayer, and even that usually doesn't last long. With the promise of acheivments, I end up playing through games multiple times, on different difficulties, and doing alot more in multiplayer. I feel better about spending money on games now.
4. Repeat #3 for emphasis, it's the best reason.
I think that's just about it, but that sounds like a good argument for me. I truely believe acheivments are one of the best features of xbox.