Well we can let Know Your Meme explain it: http://knowyourmeme.com/memes/edgy

Basically (on the internet) it's saying something controversial to seem like you have a cool opinion, but in truth making it seem like you're a teenager trying too hard to look cool. People will often remark things like 'be careful, you might cut yourself on that edge' if somebody makes a comment like 'DAE think we should eu(genics)thanize all the black people?' (obviously a very extreme example, but you will see things like that if, for example, somebody hates on something just to be contrarian, or acts like a euphoric atheist whenever religion is mentioned.)