To redeem the offer, see this guide: Apple TV+ - One Year Offer - Apple. Redeem the offer on your new Apple device in the Apple TV app. Redeeming the offer will start a subscription with monthly payments, but you start with a year free trial. Payments will begin after the trial, if you are still subscribed by then.
Do not accept the regular 7 day trial, if you believe that you qualify for the year offer.
If you qualify, and if you did as described in the guide, and you still do not get to see the year offer, then contact Apple Support. They can assist in getting you the service, if needed using activation codes.