Rewards, incentives, etc are a fundamental aspect to game design. These things provide motivation for the player to play the game. The reward could be as simple as seeing the next level, seeing where the story goes, or could be unlocking a new character, uniform or whatever. Achievements provide a universal reward system to motivate players to play. It's not going to be the ONLY thing that gets you to play, but say you're pretty happy with the game, but kind of seen most of the things the game has to offer, usually you'd stop playing it, well a lot of people would. When there's a specific achievement that can give you the motivation to play the game that much longer, the motivation can push you over the edge to keep you playing.
I'll give a personal example. I beat Perfect Dark Zero
on Secret Agent. I'm not the type of person that goes back and replays a game on the higher difficulty level, I kind of do, but usually don't think it's worth my time. Now, PDZ offers an achievement for completing the game on Perfect Agent so I have that extra amount of motivation to push me over the edge and compete it on Perfect Agent. And actually, by playing the game on Perfect Agent that has let me find out some really cool aspects of the game that I wouldn't have been aware of otherwise, which has increased my overall enjoyment of the game.