There are two separate music libraries. The iTunes Store library has songs you purchase (actually pay for) and download: you can keep them permanently.
The Apple Music library (which is slightly smaller) has songs you can stream or download (but not move or copy) as long as you have a subscription (first three months free). If you stop subscribing you will no longer be able to play the music. If you don't cancel the free trial you will automatically be charged at the end of it: your access to the songs will disappear as soon as you cancel it. If you start paying the subscription and later cancel it you will have access until the end of the current subscription month.
It rather sounds as if you had the free trial, and downloaded some songs, but then cancelled it in order not to be charged for the subscription. You can still purchase songs on an individual basis (to keep permanently) from the iTunes Store: if you want to resume access to Apple Music you can subscribe (but you won't get another free trial, of course).