What show were you trying to watch?
If you have an Apple TV+ subscription then select Apple TV+ content in the Apple TV app. (The app also has content that is not part of Apple TV+, which will have to be paid for separately.)
Select first episodes may be free to watch for non-subscribers, as a teaser of sorts. Else users need to be signed in with the Apple ID account that has the subscription for that service.
See your subscriptions overview - Apple
Unfortunately, the Apple TV app on Fire TV devices doesn’t currently support in-app financial transactions [2]. Users that want to buy, rent, or subscribe, will have to do so on a different supported device. Once the transaction is done, then users can view the paid-for content in the Apple TV app on Fire TV, as linked through their Apple ID account.
To sign in to the Apple TV app, find the in-app Settings (sprockets wheel icon), then Accounts, and enter the matching Apple ID account credentials there.
About Apple TV+ - Apple
Apple Originals - Apple TV+ Press
List of Apple TV+ original programming (episodic/series) - Wikipedia
List of Apple TV+ original films - Wikipedia
