I think most of you are hitting on a lot of little truths for why achievements/trophies exist, as well as why people care about them.
As a long time Destiny player, I have hit every trophy available to me, mostly just by playing the game. I have a large clan and assorted friends who play with me. The last 3 or so trophies are harder to get ramping up from "complete a raid", "complete a raid with only your clanmates", "Complete a heroic raid" (harder difficulty), and finally "Flawless Raider" (which is complete a raid without anyone dying). There are only two of us in the clan and assorted other 20 some players who can say they have that last one, "Flawless raider".
So right there we have a few reasons.
Because it gives people a sense of accomplishment that triggers certain centers in the brain to cause pleasure.
I think there is a huge amount of truth to this. As you progress through a game's story or campaign, you are often rewarded with low level trophies. They are easy to get and added to your collection with little to no effort on your part. Sometimes this is enough to push someone who wouldn't normally consider finishing a said game to push through and get all the easy trophies. This flows nicely into my second point:
But seriously, even though this "achievements" are set by the company, it's a way of the developer rewarding you for completing the game under their conditions, the way they've intended you to complete it.
It's also a great way to artificially lengthen a game. Sometimes when you finish a game its time to put it down. Others, you might not be ready to part with said game but have no idea what to do, other than screw around. Trophies can give you a clearer roadmap to post game or new game+ modes or bosses you haven't dealt with yet.
Which, speaking of end game, most of the harder trophies are collected for bragging rights, as others have already pointed out. Its...
...The same reason people climb Mt. Everest.
There's also self-imposed challenges (non-trophy) which could be fun, too.
To say they did it. To brag. To show off their collections. The same reason people collect... anything. There is no
real point. Its all just for fun. I would have never attempted to finish an old, outdated Destiny raid by myself, unless there was a reason. And that reason was that last trophy I hadn't gotten yet. I now get to tout it over my friends that I could do it and they haven't.
At the end of the day, trophies don't mean anything. Super Mario World didn't have trophies, yet I had just as much fun or more unlocking all the levels, beating all the castles, swearing at star world... You get my point. You can pay as much or as little attention to trophies as you like.