I can't open my songs in iTunes without the Apple Music

You can't open songs in iTunes at all. On the iPhone, iTunes is a store. You use the Music app to access your music.

What is that supposed to mean?

Any music you purchased from the iTunes Store, or some other store, or ripped from CDs, you own. Any music you download as part of an Apple Music subscription, you are just renting.

Glad it helped.