People farm because they want to max out their character, it's the same as in any MMO, you farm a certain raid (or raids) to get a specific item.
Now the main difference is that in diablo loot is randomized so there is alot less chance to get the "perfect" item for you or your build, but this again
makes it so much more rewarding when you finally get it. As a side note, i'm on act2 inferno, getting my ass kicked.. so i'm playing on my low level
barbarian while waiting for the inferno nerf (1.0.3)