I was having issues with this, too and just talked to an Apple service guy. After redeeming your 3 months free subscription, (on your computer) launch Apple TV+ and go to the top of the menu bar and go into the Account tab and enter your AppleID, (see screenshot below). This should do it. It did for me.
Originally, I clicked on the link to redeem the 3 months free subscription from Best Buy on my iPhone and it worked fine on my iPhone, but I was trying to watch Apple TV+ movies on my iMac Pro and nothing happened. I got the 'Start 7 Days Free' button and that was annoying. Make sure your computer or TV has the same AppleID that you used to redeem the 3 months free subscription. Hope this helps.