Just reached this event last night (Horde here). You get the quest chain early, it's only the 2nd quest zone area (Dragonblight, your first quest zone area is either Borean Tundrea or Howling Fjord) so you should get this quest around level 72. Not sure how it works, I completed all my Venomspite and Agmar's Hammer quests (that's the two main Horde quest hub in Dragonblight) before I got this quest. Maybe it's like Nagrand's grandmother chain, maybe simpler. Basically says you're now a veteran and to serve Younger Saurfang at the Wrathgate.

But Younger Saurfang needs permission from the Dragons, so you go to the Dragons, complete a fun dragon flying quest, and go back to Younger Saurfang. Once you turn in the quest, the cinematic plays. (and there's even an achievement to boot)

Yes, it's a cinematic, a machinima actually, it's not really an in-game cutscene. You can even find the actual .avi in your game files.

You can talk to that red dragon queen again to replay the cinematic at your leisure. Don't worry if you're on a PvP server, she emits an aura that won't allow anyone to attack. So you're safe while afk and watching lol.