The idea is that most of your controllers share something in common with each other. For example: All admin controllers need to make sure a logged in user is present and that they are an administrator. A public controller may want to load a theme for your application and load default user data, navigation links or anything else frontend related.
The problem is solved by creating a base controller - in his example its one called MY_Controller that follows the CodeIgniter naming convention and allows you to easily make other controllers that extend it. You'll also need to make a small addition to your config.php file to get the base controllers working correctly and make them able to be found.