First thing you need to understand is that there is NO such thing as a free “Apple TV” subscription. The free subscription is for Apple TV+ (plus) which is Apple’s original content streaming service and completely separate from the rest of the Apple TV App. You do not get to watch anything you find free, only the original content.
Launch the TV app on your iPhone, scroll down to the Channels section and tap the Apple TV+ icon (black icon). There you can watch any Apple TV+ content as part of your subscription.
if you are using an Apple TV steaming box, launch the Apple TV app, then see the Apple TV+ tab at the top of the screen.
iTunes movies are NOT part of the Apple TV+ subscription.