Complex sentence: The name for the magazine, "Alternative Press", was not a reference to the alternative music genre, but refered to the fanzine being an alternative to the local press that wasn't covering the music that Shea felt deserved to be heard.
Split into simpler sentences: The name for the magazine, "Alternative Press", didn't come as a result of the music it covered. At the time, The name came as a result of the fanzine being an alternative to the press that was in town; that wasn't covering the music that Shea felt, deserved to be heard.