MPAudioVideoRoutingTableViewController is a UITableViewController subclass which takes the number of available 'routes' the audio can be sent through and presents them as options to the user. In other words it is the Airplay options screen you see after pressing the Airplay button).
I believe, but have not fully tested, that the MPAudioVideoRoutingTableViewController is never used directly but rather through 2 other classes. On the iPhone MPAudioVideoRoutingViewController is shown fully screen. On the iPad the it is shown using a MPAudioVideoRoutingPopoverController in an on screen UIPopoverController.