So, perhaps I'm "old school", but I do not generally set out to get achievements. I'm also not the kind of person to assiduously work towards an "s rank" (in fact, I generally don't care about achievements unless I happen to get them on my normal playthrough). I play a game until it ends and - with the exception of multiplayer - I generally just put it back on the shelf or take it to EB Games (Canadian Gamestop).
 
However, I've noticed that these things are very important to some of the others in the community so I'm curious to know what it is that is so appealing about them (DLC and S rank, that is).

gives me something more to do once I beat the game.  If I enjoy the game, then I'll try to complete it 100%.  If the game was fun but not that great I'll beat it and probably trade it in.
 
However, I hate the games that have achievements that require you to get other people involved.  Like Dead Rising 2 makes it were you must complete the entire game with COOP.  Which is stupid because the COOP isn't that great/fun thus, that game won't get completed further after I'm done with the storyline.
 
Loving Brotherhood so I'll probably try to complete it 100%.  Which is rare for me.  Easily my number 1 or 2 game of the year next to Red Dead Redemption.  Kind of a shame that they pushed the MP so much because the singleplayer is by far the best part of the package and easily better than the 2nd game imo.

For me DLC is just a way to get back into a game I feel like playing, but don't really want to replay the story. 
 
As for Achievements, and S-Ranks... generally they entice me to do something I would never do if I simply played it my own way. (Like killing a guard with a broom, hilarious.) But at the same time if they aren't going to be fun to get I don't even bother, like grinding levels or killing large amounts of the same monster, or beat the game without getting hit. Those are not fun and don't add anything to the game except frustration. 
 
Assassin's Creed II: Brotherhood has a great set of achievements so I think I'll try to S-Rank it, plus the multiplayer is amazing, and it is unlike the more popular FPS multiplayer so it may have some serious longevity for me. Can't wait to start playing again after I finish typing this.  Arriverderci!

An incentive do play again but differently, if it wasn't for trophies I would only see 1/4 endings for example.
 
They have replaced cheat codes basically.

What's not to understand about extra content and specific challenges? They can both be arbitrary or great, depending on if the developer gives a shit and has a sense of skill vs. time investment. Not saying you should care though. It's just there for those of us who like to get deeper into any particular game.