Is my ignorance showing yet?
Yes. You appear to know about as much jQuery as I do. 
There are lots of scripts on that page (far too many, really) and I con’t tell which is creating that effect.
However, if you turn off JavaScript in your browser, next to nothing appears on that site, meaning that it’s far too dependent on JS—which should be used to enhance a site only.
So I’d steer clear of it if I were you. It’s bad news.
Strictly speaking should I be defining this bit … into a separate style sheet and linking it to the document?
Yes. The on-page CSS was just to make a demo easier. Whenever you have styles that apply to more than one page, an external style sheet is much more efficient.
And I’m guessing that image links for these boxes (block elements?) and onClick commands go in these lines in the HTML:
Yes, that’s were they go. The boxes (<div> elements, which are block level by default) are where the images and links would go. Although onClick stuff (JS) could lead you astray if you are not careful, as I said. What would you want onClick for?
Also if I did want to try and get the javascript transition effect am I looking at a something complicated that may not work cross-browser or is it a well-tested and bulletproof method?
I don’t know about the method itself, but one of the advantages of jQuery is that it is designed to work nicely in all the main browsers. It has in-built functions to smooth over the annoying differences between browsers. But that’s jQeury itself. I can’t speak for any plugins that others have written, though you’d think they would have tested them.
The code I posted was a demo only, and you would probably want something a bit more sophisticated for your actual site.