achievments show up in your gamer score, which really dont mean anything, but people with a lot of gamer points are usually heavy gamers, and sometimes pretty good. you can compare you acheivements with friends, and they are viewable by anyone who wants to see them. they are like little acolades youve earned in playing your games
in addition, i feel like it makes playing games more phone, although achievements dont treally do anything, it does make playing the game that much more fun. something to strive for. for example, when i played angry birds on android, all i wanted to do was beat every level. when playing it on wp7, i wanted to beat every level with 3 stars so i can unlock the acheivement. and so my friends could see that i accomplished this.