If you create a game where the main character just jumps on the spot whenever you click the mouse button, it's generally not going to be very popular. You may get a few desultory clicks, and the 'game' will get ignored or uninstalled without further ado.
Throw in an achievement for a thousand clicks, however, and a good percentage of people will do it - just to get the achievement. It will still be a crap game, but now you 'get' something for your effort.
Achievements don't really add anything to a game other than to go "WOO!" at you whenever you do something that the developers consider to be notable - but they do seem to promote replayability, potentially getting the player to do somethng they normally wouldn't consider to be worth doing.
How many people would set themselves a goal of making that pixelated little character jump a thousand times, if there was no more reward than yourself going "WOO!" or shouting "Achievement Unlocked!"? Get the game developer to offer the achievement, however, and suddenly it's a task worth doing properly.
If you make money from people playing your game - through exposure to advertising or whatever form of monetisation - then achievements are a godsend. They cost practically nothing to implement from a complexity level, and they can massively boost your player longevity the same way that "Step 5: Rinse and Repeat" singlehandedly saved the shampoo industry.
They allow you to develop less content, because players will be happier to repeat parts of a game if they're "working towards" an achievement - and this is certainly a trend I've seen a lot of just lately. Diablo3 springs immediately to mind. World of Warcraft is another. A bit of a blizzardy theme there, but they seem to have embraced the concept whole-heartedly.
However - why can't we come up with our own achievements? I know I used to set myself little goals in games. In Asheron's Call it was to get my "Run" skill up to ludicrous levels, because that was the best way to avoid being killed by gangs of opposing players. In Civilisation, it was winning without using any diplomacy at all - or seeing how far I could get in NetHack without actually killing anything.
Now if you don't get an achievement, it's just not worth doing, apparently.
I'm not saying it's a bad thing at all. Frankly, I like them, because when a game forces you to repeat content (and most of my preferred MMO genre do, when you get down to it) it's nice to have the drudgery occasionally broken up with a congratulatory message telling you how many times you've jumped on the spot.
I'm just a bit cynical, because I know deep down in my crusty little gamer soul that it's usually a cheap gimmick someone's either using to squish a bit more money out of you, or to disguise the fact that they only developed five levels in their game instead of ten.
"Are you not entertained? Are you not entertained? Is this not why you are here?"