Actually looking at the two, it might be a script conflict. Your site has a lot more scripts running than the demo site. And thinking about it, if you change that css I suggested then you probably won't have the background-image do that nice slide over effect.
How did I figure that out?
Basically Firebug, if you use Firefox you can install an add-on called Firebug, which is immensely helpful for troubleshooting these kinds of things. Remy Sharp has a nice video tutorial that will help explain how to use it better: http://jqueryfordesigners.com/debugging-tools/
If you use Safari or Chrome, they have a built in developer tool, it's in the menu but it should load if you press F12 on your keyboard.
Another immensely helpful Firefox add-on is the called "Web Developer" by Chris Pederick. Can be found here: http://chrispederick.com/work/web-developer/
Anyway, back to the problem. Do you have a lot of plugins installed that didn't come with your theme? If you have a test environment, try turning all your plugins off and see if the animation you want to happen comes back. If so, then it's probably a script conflict. Try turning the plugins on one at a time and check the animation after each one. This would narrow it down to one plugin (hopefully) and you can go in and see what scripts that plugin is using and see if there is a conflict.
The link to the theme demo (not your site) has 12 scripts running, your site has 20 scripts running. So it's probably something to do with that. The problem is based on a class "active" which is a rather common word to use to describe the state of something, unfortunately this can lead to issues if two scripts are telling the browser to do something when the class "active" is added to an element.
Try the plugin thing, oh and if some plugins came with your theme, try leaving them on or turning them on first. Try to get back to an unaltered version of your theme. If it starts working, then turn the others on one by one and see if it stops (each time you turn one on).
Let me know if that returns any results.