This is possibly a stupid question so apologies in advance, however, once you have downloaded a game from your library what do you do with it? I've only ever bought games through steam where it does all that stuff for you, I'm not sure what I'm doing with the .zip file now I have it.

Simply right-click the ZIP file in your file manager, and you'll be given the option to unpack the archive. You'll get a folder, that should contain an executable. Simply run it. There are variations on the theme sometimes.

If that sounds too complicated, install the Itch app and it will take care of things for you.