I first learned about it from the first chapter of Lion and Vergne's book "The Weil representation, Maslov index and theta series", which is really much more down to earth than its title might suggest.
EDIT : It maybe should be emphasized that I am not a representation theorist, and when I read the above book I knew far less about the subject than I do now (which isn't a lot). This seems to me to be a positive thing -- I found the book pretty easy going despite not knowing all that much when I looked at it.