Why do I have to purchase iTunes Match or Apple Music to listen to CDs I already own??

Can you please post a screenshot of the error message that you're seeing? iTunes Match and Apple Music are both library syncing services that you can use to access your library (and with Apple Music, additional content) across multiple computers and devices. Neither are required to enjoy music with the Music app. If you have an iPhone/iPod/iPad this can be synced over a cable or wirelessly when on the same network without using either subscription service.

tt2

Check on the Music > Settings > General tab that you don't have Sync Library checked. I don't think it should even appear if you don't have a subscription, but right now it is the only thing I can think of that could trigger such a message.

tt2

Have you previously had a subscription which you have cancelled? The red download arrow on the right of your screenshot suggests these tracks are in the cloud rather than stored locally. If they were purchases they they should stream or download, but the pop-up does seem to indicate you are looking at Apple Music content.

tt2

Be sure to play the music from your library. No subscription required.