apple tv not playing apple music library

The Music app on the Apple TV box will access content from the online iCloud Music Library, i.e. from an Apple Music subscription, from an iTunes Match subscription, or from iTunes Music Store purchases.

For access to music in the library on a networked computer (iTunes app or Music app), use Home Sharing and the Computers app instead.

AirPlay should also be available to you. Use AirPlay to stream audio - Apple Support