This implementation is taken from TrainLord as it runs on web (flash), android and iOS (AIR) platforms. Facebook’s authentication flows are based on the OAuth 2.0 protocol. Its recommended you have general undesrtanding of OAuth protocol prior reading this article.
If you run services on your own backend (like TrainLord does), the main point of whole thing is to securely pair a facebook id and user session. So later, whenever user does a request to your backend it automatically recoginzes the facebook id. In my case I used id and name to create an unique user record in my database so I can save some more user related data (score, etc.).