The module includes Zend library files so folks who don't have Zend should still be able to use this.
The module uses Zend 1.9+ - supported by most Apache servers. This has been tested on a LAMP machine (Linux/Apache) but in some cases - Windows servers, perhaps, or if your host doesn't allow PHP scripts to use the ini_set function - it may be necessary to manually set the include_path directive in your php.ini to something like 'include_path = ..:/usr/lib/php:/home/account/public_html/modules/mod_randomyoutube/Zend/library'.
The above assumes your php.ini include_path is /usr/lib/php, but that will be different for every server, so check phpinfo for that. In the Joomla backend, go to Help > System Info, PHP Information.
As clearly indicated, this module requires Zend. This is because in order to use the YouTube API, Zend is required. If it could be done in a simpler way, I would. But as it stands this is the only module that allows you to show a random YouTube video from a specific YouTube user account. Why is it the only one? Because you have to use the YouTube API to achieve this, and that requires Zend when using Linux/Apache/MySQL/PHP.
I don't believe that killed the rest of your site though. If you uninstalled the module, it should be back to normal.