An anonymous inner class can come useful when making an instance of an object which certain "extras" such as overloading methods, without having to actually subclass a class.

I tend to use it as a shortcut for attaching an event listener:

button.addActionListener(new ActionListener() {
    public void actionPerformed(ActionEvent e)
    {
        // do something.
    }
});

Using this method makes coding a little bit quicker, as I don't need to make an extra class that implements ActionListener - I can just instantiate an anonymous inner class without actually making a separate class.

I only use this technique for "quick and dirty" tasks where making an entire class feels unnecessary. Having multiple anonymous inner classes that do exactly the same thing should be refactored to an actual class, be it an inner class or a separate class.