The site generates a .mobileconfig file, which is an Apple plist file in XML format. The API is public, and easily reverse-engineered from other .mobileconfig files.
The "app" that gets installed is actually a webclip, not a native iOS application. They embedded the icon as a base64 string inside the .mobileconfig.
The webclip opens mobile Safari in fullscreen, with a link in the format of "http://mocava.de/ios/parental-control/profiles/UNIQUE_ID/par..., where UNIQUE_ID ties back to the choices you made on the setup screen of their website. This will allow Mocava to track usage stats, which seems like a fair trade-off for a free service.
You could roll your own version of this using Apple's Configurator application, however Mocava has done a decent job of streamlining the setup and installation.