Do I have to buy the game again?
No, unfortunately itch.io is a bit confusing. Check your library for the file. Max Cheerum included a very cool guide above
Check your library on your profile. Click on the games download button, it should lead to a page where you can get all the versions they have placed in there. Then just go to the file you wanna download.
For further understanding why you have to do those steps, is because you cant download the file directly from here, but from your library folder, or from the main page of In Heat.