How do I access free Apple TV after I have purchased a new MacBook Air?

The offer is for Apple TV+ (note the Plus), which is different from ‘Apple TV’.

Redeem the offer on your new eligible Apple device, before trying to use the service on other devices.

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

The offer is for users than haven’t had an extended free trial for Apple TV+ before: One offer per user.

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

If you go to Settings, there should be a "subscriptions" page which allows you to redeem free access to things like that.