How can I get Amazon music on my iPhone?

Download Amazon Music from the App Store and sign in with your Amazon account.

If you are trying to do this in the iPhone in your tag line, then you can't download it as it requires iOS 13.

https://apps.apple.com/us/app/amazon-music-songs-podcasts/id510855668

If you had dowloaded the App before you can try to get it form your previous purchases and see if you are offered a compatible version: link-> Redownload apps and games from Apple - Apple Support

See if you can get to it via the website - https://music.amazon.com