First of all, you need to download your music if you want to play it offline.
To download a song, album, or playlist:
Tap after adding music.
Important: The Sync Library option must be turned on (go to Settings > Music, then turn on Sync Library).
It's also convenient to have your music downloaded always:
Go to Settings > Music, then turn on Automatic Downloads.
If you'd like to see the download progress:
While on the Library screen, tap Downloaded Music, then tap Downloading.