Why do I have to be connected to listen to music I own?
Why does iTunes want me to connect to the internet to listen to music I own outright?
I assume it has something to do with my apple music subscription.
iPhone 6s Plus, iOS 9.0.2
It does. When you signed up for Apple Music you signed up for a streaming music system. You have to create Offline Playlists to listen to music offline. This support article should help:
Add music from the Apple Music catalog to your library - Apple Support
Cheers,
GB