How do you get closed captions watching Amazon Prime movies through Apple TV?

Actually, it's easier to just go to your Apple TV Settings -> Video and Audio -> and make sure Subtitle Language is turned ON. If it's already set to English, turn it OFF, exit and then go back and turn Subtitle Language back to English. It's now working for me with the Amazon Prime Video app on Apple TV. Not sure what changed but it's now working for Amazon app on Apple TV for closed caption. Just need to set it to English or language of your choice in Apple TV Settings.

Check out the instructions from BigToeJoe in the thread Re: Amazon Prime app Subtitles missing no (CC)

It wasn't working for me initially, but it seems like they've silently pushed out an updated version of the app for Apple TV 3rd Generation :

https://discussions.apple.com/thread/8190740?answerId=32731769022#32731769022

Applet111

Which Apple TV do you have?

Skymonkey2000

Gen 3