The service is extensively described on its main instance, http://auth.osmz.ru/.
Copy all files, including .htaccess, to a server directory. It would be best to place it at the domain level, or else you'll have to tweak links in web.php.
Register the application in OpenStreetMap and put OAuth and database settings into config.php. Then run createdb.php and immediately after that delete it.
An example usage of the proxy service is published in the gh-pages branch.
Copy a file you understand in lang directory to your language name, for example, it.php. And translate it. Don't forget to create pull request.
ZeroClipboard was written by Jon Rohan and published under MIT license.
OSM Auth Proxy was written by Ilya Zverev and published under WTFPL license.