To prove my point, I would like to direct you to my old friend, the PS2. Now, I was an avid gamer, but as games developed more graphically, game kinda ended and then that was it.
Period. I beat the story, got the best weapons. In the later stages, few games had any real EXTRAS, something to sink my teeth into after I beat the game.
Now enter Ratchet and Clank, one of my favorite series. A way that all of the games caught my attention for hundreds of hours was the vast ammount of extras and "Skill Points".
You see, I could beat the campaign 100 times, but wheres the challenge in that? Try beating 100 rounds of the Arena with only your wrench, and NOW we got ourselves a challenge. It was fun to do, not mandatory, and kept my playing.
Achievements are that, exactly. They're supposed to be extras awarded for playing games. It gives gamers something to go back and do after they beat the initial story in the game. Essentially, its fluff. Fun fluff. Achievements ensure that gamers continue to play a game long after they have beaten the story and/or grown 'meh' with the multiplayer. Its something fun to work towards.
Gamers come up with unique challenges for games they play all the time; an example is a scramble challenge in the Pokemon games (which adds unique twists to the standard teams). Achievements are like that, but added by the developers. Its just fun. Mindless, addictive, fun.