Screen Mirroring

For playback on any device that differs from the device on which it was downloaded, the playback software wants to verify if you are authorized to play the file, by contacting a remote server. You can play it on the iPhone offline, but not stream to a different device while offline.

The authorization for offline playback was only for the device on which it was it was downloaded so it has to contact the server again to verify one's authorization before it can be played back elsewhere. This won't apply for non commercial/DRM content, and some services may block downloads from airplay altogether.