They can often make them look very authentic, and Apple doesn't typically send such emails. Even if they had started to here, as a reminder to claim the offer, the watch doesn't qualify so that would seem very suspect. Anyway, if you do happen to purchase a qualifying product (iPhone, iPad, Apple TV, Mac) then I will list the steps below for claiming the offer.
- Turn on your new Apple device and sign in with your Apple ID.
- Open the Apple TV app. Make sure your device is running the latest version of iOS, iPadOS, tvOS or macOS.
- The offer should be presented immediately after launching the app. If not, go to the Apple TV+ channel page where you’ll see the offer.
- Tap Enjoy 1 Year Free.