What is this?
This is called ‘Progressive Web Application’. It’s not a true windows or android app but a browser-based one. Though it’s designed to work locally on your device in offline mode with minimal UI wrappers, you can have a shortcut for it, uninstall it and it acts mostly like a real application on your device.
How to install?
After clicking the ‘Play’ button you’ll be redirected to the app page, where you can install it on your device. Some browsers will invite you to install it automatically, some have the installation button around the address bar, some hide it in their menu (I collected a few examples of how this might look). Wait for it to be installed… then it will appear in your list of installed applications and might even get a desktop shortcut (this depends on your settings).

UPD: The latest version should show the ‘install’ button over the game screen, so you should not look for it in menus anymore.
How to update?
Once there is a game update, the ‘server’ version will be updated as well. However the actual Patreon subscription is required to access it. You can return to this page and reinstall your game the same way: press ‘Play’, follow the link and wait until your game is updated in the browser. After that the new version will be available for offline play again.
UPD: since v1.8.0.5 once the version update is available the app will show the ‘<‘ badge right side. Click it for quick access to the ‘download’ page.
Attention!
The whole ‘app’ thing is being tested at the moment. There are some known issues and I’m working on them. Though you might catch something I don’t know about – feel free to share it!
Troubleshooting
“I get numerous visual glitches when playing”
Try to restart the game.
“I ran the app and see only the black screen and a small text”
Reinstall the application. Probably the game wasn’t cached properly OR the cache was cleaned (though this should not happen theoretically).
Updating from v1.8.0.4
App version 1.8.0.4 contains few errors, so updating from it to later versions doesn’t work properly. The key feature to recognize it, it has the ‘full screen’ button right side, it was moved to the left in the next release.
When updating from it, it will probably say about error when caching OR it might invite you to click to refresh the page infinitely. In both cases please close the tab (or all tabs if you have multiple tabs with the app opened) and re-open it in a new tab for changes to apply.
If this doesn’t help (and you still get the ‘full screen’ button right side), try to clear the cache for the app/site and refresh the page. In desktop chrome this requires to refresh the page with Ctrl+F5. In mobile chrome open the page, tap the three-dot menu > Info icon (i) > Cookies and site data, and tap the Delete. For other platforms/browsers google for ‘android chrome clear cache for one site’ (insert your platform and browser) to get the detailed instructions.