Hello,
The below article give instruction on downloading songs onto your device. I am not sure if the Android version of the app has this setting but, on an iPhone if you go to Settings > Music there is a toggle for Automatic Downloads. If you have that and enable it the songs may download on their own in the background.
Add and download music from Apple Music - Apple Support
"Download music from Apple Music
After you add music to your library, you can download it to your device to listen to offline. Just follow the steps below for your device.
How to download music from Apple Music on your iPhone, iPad, or Android
- Open the Apple Music app.
- Add a song, album, or playlist to your library.
- Touch and hold the song, album, or playlist that you added from Apple Music.
- Tap Download."
I hope this helps.