Laravel Repositories
Repositories, like in the provided tutorial, aren't necessary a Laravel concept. Rather, they're a form of IoC injection that is possible with Laravel. Any object that might similarly be injected doesn't mean it's a repository. See the video for a good example from Taylor Otwell, which happens to use a "repository" as well: http://vimeo.com/53029232.
In this example, the repository abstracts where the data came from that is passed to the controller. As long as the data passed implements the specified interface, the controller can "blissfully" make use of the interface's defined methods without worry about where the data initially came from. This allows switching the initial source of the data without breaking your controller. You could pull the data from a file, a database, an outside API, a mock object, or just some arbitrary array. Basically, the controller doesn't need to gather the data represented by the repository. It can just receive and use.
Repositories help to keep your controller clean and make reusable code. Functions in repositories can be accessed in one or more controllers, repositories, etc.
Also, all your backend related logic (like fetching data from database or external calls) can be added in repositories to make the logical separation.
One of the main uses of repositories is to create different binding (using interfaces to define your functions and with the help of the app, bind different implementations of the function as per need). For example, two separate repositories (implementing the parent repository/interface) handling database and files for backend data.
The main reason you use repository pattern is to be able to easily change your data source. For example, in my experience the most common change is adding a caching layer, because the repository implements an interface all you need to do is build the new object implementing the same interface with the new methods handling cache and change the binding.
In addition to the other answers here, it's worth pointing out that repositories used when used in Laravel can add an extra level of expressiveness. Take for example the following:
$users = User::whereHas("role", function($q) {
$q->where('name', 'moderator');
}, '<', 1)->get();
The code is difficult to read and awkward to look at. It can be encapsulated in a repository method, and demonstrate much clearer code intent:
$users = $userRepository->getUsersWhoAreNotModerators();
This is also achievable using eloquent 'query scopes' but I think using a repository is superior, as is adheres much better to the single responsibility principal, and is doable regardless of whether you are using Eloquent.