My problem is the same problem that the person in this Github issue had:
https://github.com/itchio/itch/issues/1018
It appears that that has already been fixed, so there must be a different reason for my problem as I am running the latest build. I have been using the Itch application for Windows for quite some time, but just made my first purchase today. I am logged into the correct account and the purchased items show up in my library. However, when I click to install them, they show "itch.io API error: you must buy this game to download" on the download screen.