You need to redeem your Steam key. It says this on both the store page and the download page after purchase.
You have to redeem the Steam key in order for it to authenticate your ownership of the game. Even if you're not using the Steam version. "The Itch.io version of the game has custom DRM that requires that you own the game on Steam. It is non-intrusive and simply verifies ownership of the game. Please redeem your Steam key before playing." Perhaps that could've been included in the piracy message though.