Hi chronixx. Welcome to the forums.
This can be done with CSS alone, but it looks like you are wanting to do it the jQuery instead. If so, we can move this to the JS forum. At the moment, the JS tells the browser to "show" the box, but doesn't tell it to hide it again afterwards.
The problem with using JS for this is that people without JS won't have access to the menu. You also have to consider people on mobiles, who can't hover.