The offer has to be redeemed by the user. The subscription is not automatic.
Read here how to redeem: Apple TV+ offer - Apple
Read here about the conditions: Apple TV+ offer - Terms and conditions - Apple
Read here if you have trouble redeeming: If you can't redeem your promotional trial of Apple TV+ - Apple Support
Note that the offer length differs for devices activated (first use) before and after July 1st.
If you do have the subscription for Apple TV+ already, then this service would be available in the Apple TV app on any of your devices, when signed in with the Apple ID account that has this subscription. See your subscriptions overview - Apple
Note that for Macs, users have to additionally sign in to the TV app (menu Account), for the app to recognize the account. Note that for third party devices, users have to sign in to the Apple TV app (sprockets wheel icon for in-app Settings), for the app to recognize the account.
The offer is for Apple TV+ (Apple Originals), and other items in the Apple TV app are not included in this subscription. Notably iTunes Store offerings for Buy or Rent will have to be payed for separately.