How can I create a similar page transition on my website?

Hire a professional that will cost a lot, that’s how they probably did it.

You could also just right click and view source and try and figure it out.

@freckoes Welcome to Fluther. You have excellent tastes in websites. That’s an elegant presentation, to be sure.

A quick look at the scripting in the head tags suggests that it makes very extensive use of the jquery JavaScript library. Like @XOIIO said, this would be beyond the reach of all but the strongest developers, and most likely took a good deal of time for a large team to develop.

Start with CSS transforms. There are a lot of examples and tutorials.