Not sure if I'm entirely correct, but "emo" is short form for "emotional", and people who SI are often called "emo" because they tend to have more emotions, or emotional trouble than most.

Emo is also a style of music and clothing.

Really, everybody is "emo" to some extent because we all have emotions. It's just that the stereotypical "emo" is generally associated with cutting, the colour black, mascara, etc.