To demonstrate this, I've put together probably the most boring proof-of-concept ever. You'll have to use your imagination to see how this technique can be applied in much more creative and better-looking ways.
First, I created a simple animated GIF of a white bar moving left to right over a transparent background. I actually used Flash to create this. In fact, you could take a Flash interface, and publish the different pieces to animated GIFs in order to eliminate the need to use Flash. Anyway, here is my boring GIF on a black background, so that you can see what's happening:
Now, I'm going to use this together with CSS to create an animating hover effect. Here is the CSS and HTML I will use:
I'm using an empty span inside the link to act as a placeholder for the animated GIF. When the mouse is hovering over the link, the span comes to life by expanding to an animated GIF which appears above the text. Since the background of the GIF is transparent, and the background of the text is white, the effect is that of a "shine" washing over the text.