The sorta free, sorta not Call of Duty Elite community and website service has a lot of moving parts. Here's a look at one of those: The game's Compete features.
I think justification for the CoD Elite service can be summarized in two words: "bragging rights."
If you're in a clan, do clan stuff, want some sort of proof to show off to friends or strangers that you're 1337 in something, then it's for you.
If you just care about playing with your friends, playing for fun by yourself, enjoying the game for itself, then Elite is not necessary.
(I'm just talking about the online services, not the access to DLCs which I think is why a lot of people will be subscribing.)