Using the Iterator interface allows any class that implements its methods to act as iterators. The notion of an interface in Java is to have, in a way, a contractual obligation to provide certain functionalities in a class that implements the interface, to act in a way that is required by the interface. Since the contractual obligations must be met in order to be a valid class, other classes which see the class implements the interface and thus reassured to know that the class will have those certain functionalities.
In this example, rather than implement the methods (hasNext(), next(), remove()) in the LinkedList class itself, the LinkedList class will declare that it implements the Iterator interface, so others know that the LinkedList can be used as an iterator. In turn, the LinkedList class will implement the methods from the Iterator interface (such as hasNext()), so it can function like an iterator.
In other words, implementing an interface is a object-oriented programming notion to let others know that a certain class has what it takes to be what it claims to be.
This notion is enforced by having methods that must be implemented by a class that implements the interface. This makes sure that other classes that want to use the class that implements the Iterator interface that it will indeed have methods that Iterators should have, such as hasNext().
Also, it should be noted that since Java does not have multiple inheritance, the use of interface can be used to emulate that feature. By implementing multiple interfaces, one can have a class that is a subclass to inherit some features, yet also "inherit" the features of another by implementing an interface. One example would be, if I wanted to have a subclass of the LinkedList class called ReversibleLinkedList which could iterate in reverse order, I may create an interface called ReverseIterator and enforce that it provide a previous() method. Since the LinkedList already implements Iterator, the new reversible list would have implemented both the Iterator and ReverseIterator interfaces.
You can read more about interfaces from What is an Interface? from The Java Tutorial from Sun.