How do I get my free Apple TV from my new device, nothing pops up when I open the app

… my new device …

And what new device would that be?

For the offer from Apple: Verify that you are eligible. If you have enjoyed an extended free trial before, then you can’t have another one. Redeem within the first 90 days on your new qualifying Apple device, in the TV app. Find the “Enjoy 3 Months Free” redeem button there.

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 promotion for an Apple subscription - Apple Support

↳ Step 5 says to contact Apple if the prior steps didn’t bring a solution. They can assist.

Note that for Macs, users have to additionally sign in to the TV app (menu Account), for the redeem option to appear.

You’re welcome.