The short answer is that it is a subculture centered around hardcore music. People who are straight edge do not smoke, do drugs or consume alcohol. There are no dietary or religious beliefs tied to straight edge contrary to media coverage.
The long answer requires a bit of a history lesson
In the late 1970s and early 1980s a group of bands and kids began something that grew into a movement. It would become a movement that would outlive many of their involvement in the music and the scene. They didn't know what they were doing. They knew they didn't like what was going on around them, the self destruction, the self hatred, the pain and suffering caused by the punk mentality. The fuck you and fuck the world attitude didn't make sense to them. They took a symbol that was originally used to identify them as being too young to drink so th
at the bartenders would know not to serve