Simply hit “Request” and according to the FB, you will be receiving an invite soon. Details as to whether those invited can dole out invites of their own like in the old Gmail days remains to be seen.
You could also try to score invites by following FBeMail on Twitter, here, or @tipeveryday but no promises from us. You probably have a better chance hustling your friend of a friend who used to sleep with the old Exeter roommate of the friend who knows someone who may know someone who works at The FB. Or you can just embrace the queue.