Okay, that is very strange. Have you tried clicking the "Download not starting?" link in the bottom right?
I haven't experienced anything like this before, so it might be something with Itch's side. I can see from our analytics that the files *are* being downloaded by people, so it seems like it's specific to you, for some reason. Maybe try disabling any adblockers or anti-malware, since those can sometimes block popups? Other than that, I'm afraid the best I can do is suggest that you contact Itch support.
I'm very sorry about the inconvenience.