Note: this all changed in Java8.
What's happening is this:
addListener() takes one parameter, an anonymous class. Its type is
ClickListener which I believe is an interface, so you have to implement it. It has one method with the signature of
public void clicked(InputEvent event, float x, float y) which you implement.
Java pre-8 didn't do anonymous methods, so the best is could do was an anonymous class. This idiosyncratic syntax allows you a shorter way than creating an inner class, implementing ClickListener and its method, instantiating an object of that type, and then passing the reference to that object to addListener.
Edit: when I started this post, Ulf hadn't posted yet.