The video starts off with some narration by Mr. Fry on the "My Moon", "InfoMoon" and "LittleBIGPlanet Story Mode" options and then progresses to a quick run through of the "Castle Climb Challenge" level. (If you'd like to bypass the narration and just see my run, the actual level begins around the 1:36 mark of this video.)
I'm actually really proud of this recording as it's very clean and smooth. Upon review, I can see that it's not quite optimal and that a number of small improvements could be made, but I still think it's pretty decent.
Also, I really enjoy the music that plays while you are in the "InfoMoon" menu. Very... chipper or something.