Apple Music on iPhone is locked to only allow downloading songs from the Apple Music Store via mobile connections. You can always simply steam the songs and albums from this site while your phone is connected to wifi or cellular data. However, you can also download songs from this site and add them permanently to your Apple Music library on your phone, which makes them accessible on your phone, at any time, whether you are connected or not.
In order to do this, you will need both your Mac desktop (or notebook) computer and your iPhone, as well as a lightning USB cable to connect the two.
From your Mac Desktop computer
Step 1. Download the songs from this site to your Mac Desktop Computer
From this site in your browser on your desktop computer, download the desired songs or albums by clicking the "Download" button next to each tune, or the "Download Album" button at the top of the album page.
This will download a zip file to the 'Downloads' folder on your desktop Mac.
When the download has competed, navigate to your 'Downloads' folder in Finder and double click on the zip file to extract the songs.
Step 2. Open up Apple Music on your desktop computer
Click on the Spotlight icon on the upper right of your desktop
In the window that pops up in the center of your screen, type "Music" and select "music.app" from the popup list that appears.
Step 3. Add the downloaded songs to your Apple Music Library
When Apple Music opens, simply drag and drop the songs from your download folder onto to the Apple Music window, and they will be added to your music library.
After you complete that step, you can delete the files in your downloads folder as Music makes its own copies of the files.
The new songs will automatically be grouped by the album they were associated with on this site. You can reorganize this within Music as you want.
Step 4. Copy the songs to your iPhone
Use a lightning USB cable to connect your iPhone to a USB port on your desktop Mac.
Once connected, your phone will appear in the menu on the left side of Apple Music.
Simply drag and drop the desired songs from the list on the right to your phone entry on the left side menu and those songs will be copied to your iPhone music library.
When you are done, eject your phone by clicking on the up arrow next to its listing, and unplug the cable.
The songs will now be permanently added to your iPhone music library where you can group and manage them as desired, and play them any time, whether the phone is connected to wifi or cellular or not.