Did you click on the app and extract all to a new folder and then run the app?
No, if you've got the latest version of Adobe Air you should be fine but I meant for the game itself, click download then in the folder click on the game name/application and choose "extract all" if you're running on Windows. Then choose a folder to extract to. That should open up when it's done and the 'application' will look like an actual icon and you can click that (in the folder you downloaded to) and run it from there. I had the error you're talking about happen to me too but this fixed it for me.
You shouldn't need to install anything related to AIR to run the Mac and Windows versions. The Linux version, you might - but I'm not sure how to get that running (which is why it says unsupported, unfortunately). Still, you might be able to get advice if you hop on our discord server - http://discordapp.com/analgesic