Thanks! The DLC is in the form of different playable maps - they should show up when you start a new game and see a "Choose your Kingdom" screen.
Ah, interesting... I had not tried the itch.io app. I see - it seems to "install" the game installer (which you then run to actually install the game), but it doesn't know about the DLC packs. The two DLC packs (downloadable through the itch.io website) are separate installers that you need to run. They'll try to install the DLC in the location where your game got installed. And you'll need to have the full game unlocked with the license key, if you haven't already. Sorry for the confusion - the itch.io app does weird stuff in this case. :-/