Click on the "downloads" button, but move the cursor off it and onto the "store" button before releasing the mouse button.
Leave the cursor on the "store" button.
Here's the cool thing:
The "store" animation (with Homestar saying "Awwiiight!" and his head floating off to the right) will be triggered without canceling the "downloads" animation, so when the "Edgar the Virus Hunter" startup screen appears, Homestar's head will still be visible over on the right!
...Well okay, I guess the cool thing is not so much the thing itself as the fact that I discovered it by accident all on my own. So um yeah.
Edit: Cool, that trick works with all the buttons! You can get all kinds of strange effects by going to any Homestar Runner main page (accessible by the li'l number buttons in the lower left corner) and doing the mouseon-click-mouseoff-release trick on the buttons.