To activate a game on itch.io, simply copy and paste the URL you obtain when revealing the key into your browser. Log in to your itch.io account or register if you don't have one, and after that, click on the "Link my account" button to redeem the game. Don't forget to mark your URL as used on Groupees once you've linked the game to your account.
If the download link redirects you to the game's itch.io page, please check if you've copy and pasted the link properly, and after verifying that, contact support if it still doesn't work.