It boosts the online community, having members comparing the trophies, ranks and sharing their experiences on particular tasks that were required to get the trophy.
It helps the members working together to get the hardest trophies. We can see dedicated web pages specifically for that.
It adds to the replayability, making you play longer to complete specific tasks that you normally wouldn't do, giving only more reason to play it, spending more hours with the games, increasing the experience.
If it wasn't for this, I probably wouldn't have bothered, for example, with finding all the relics on the Uncharted games, thus not fully exploring all the possibilities the scenary gave me, or missing some amazing fights while doing the MK Ladder on the hardest difficulty without dying or finding secret characters. Or simply to replay the game to get what we previously missed.
Personally, I love seeing the icon in the top of the screen whenever I get a trophy, I feel acomplished. It is satisfying to me.
Fortunately, I learned to like it, but if one simply didn't learn to care for that, then there's not much that can be done.