MPMoviePlayerController
does not support the playback of YouTube SWF (Flash) video, period.
That app you are mentioning actually plays progressively downloaded files in MP4 format which YouTube also offers for some of its content. This actually is a violation of Apple's guidelines as it will (and does) exceed the maximum amount of progressive download per app per timeframe. I am surprised it got through the iTunes approval.
Warning: iOS apps submitted for distribution in the App Store must
conform to these requirements. If your app delivers video over
cellular networks, and the video exceeds either 10 minutes duration or
5 MB of data in a five minute period, you are required to use HTTP
Live Streaming. (Progressive download may be used for smaller clips.)
If your app uses HTTP Live Streaming over cellular networks, you are
required to provide at least one stream at 64 Kbps or lower bandwidth
(the low-bandwidth stream may be audio-only or audio with a still
image).
These requirements apply to iOS apps submitted for distribution in the
App Store for use on Apple products. Non-compliant apps may be
rejected or removed, at the discretion of Apple.
So your task boils down to the question on how to get the MP4 URL of a video offered through YouTube. That part is really tricky and nicely solved by Deja. Just use a packet sniffer and you will see that it actually creates a local server that feeds MPMoviePlayerController.