Apps, Extensions and scripts cannot be added from this website

You can now also just drag-and-drop the .crx file onto the chrome://extensions page and it will install, too.

Edit 2019-01-10: Make sure to enable Developer mode for extensions. If this was not enabled, refresh the chrome://extensions page after enabling it.

Edit 2018-08-24: This works on Windows, too.

Edit: This doesn't work on Windows. Verified on OSX and ChromeOS. From what I've read, this will work on Linux, too.~

Instead of opening the file, you have to: extract it (crx are zip files), then visit chrome://extensions, enable developer mode, and load unpacked extension.

The reason for this is: lots of people try to abuse chrome apps/extensions to install nasty applications to your browser. So Chrome tries to make it impossible to automatically side-load these apps without you making sure that is really what you want to do.

So.. typical caveats of installing software apply when using this approach.