On your computer, you can use the Amazon Music software to download the music on your PC. You can play the music directly in the Amazon Music software.
You can set the software up to automatically import the music that you have downloaded into iTunes (if that is your desire).
From there, you can copy it to your iPhone (or iPod or iPad, etc). And it will show up in the Music app on your iPhone).
Alternately (as suggested above), you can use the Amazon Music app on your iPhone/iPad to download the music. Be aware that music downloaded directly in the Amazon Music app on your phone will not show up in the native Music app on your iPhone.