If you have an Amazon Echo and use Amazon Music, you'll know full well that you can play all your music using your smart speaker. But you still have to use voice commands to sort through everything, which can be a pain. That's presumably why Amazon has just launched Alexa Cast to let you do the controlling from your phone.
It's just like using Google Chromecast, and the name similarities are surely on the radar of Google's copyright lawyers already. The principle is the same too, you can connect the Amazon Music iOS or Android app to your local Echo speaker by tapping the Alexa cast icon. From there the Echo will begin playing music as directed by the controls on your phone (or tablet, I guess).
And that's all there is to it, really. It'll be nice for you if you actually use Amazon Music, be it a Music Unlimited subscription or the Prime Music perk, since it's one less feature you don't get with Apple, Spotify, or Google.