That’s ajax iirc, it’s built into themes and sites so that they do live searches without the need for refreshing. The animation is built into the ajax code
eg: http://www.blogthe.net/ type in the search bar, wait two seconds, it comes up on the right loading and loads wihtout ever needing to refresh
I’d love to help, but i’m an absolute newbie to ajax and javascript. i’d recommend using prebuilt ajax search templates as a start and adapting them to your needs, i learned half of all the php i know like that.