The itch client doesn't seem to have support for NES ROM files (*.nes). I haven't tried other ROM types.
Currently the install just silently fails, and does not put anything on the PC other than an empty folder.
In an ideal world, there would be some sort of embedded emulator, or would attempt to open the file with the associated program (as if I double clicked the file) but I would settle for just installing the latest version of the rom file, and opening the folder containing it (this is similar to how PDF files work, I believe).
Here is a free NES game you can use to test if needed: https://mhughson.itch.io/from-below