I dont disagree with the idea of an A.I. Intelligence explosion, but I do think their argument is the wrong one. It is a fairly common idea though, especially from semi-laypeople, so I'm not surprised to have seen it in their video.
Everything else in the video seemed roughly accurate, though.
http://youtu.be/bTMS9y8OVuY
I think this is a nice introduction to Transhumanism, inspired by the style of many well known Youtube educators. Given how much LessWrong likes these ideas, I thought it was worth sharing.
The group also has a Kickstarter here to fund an entire series of videos of this kind. I think they deserve to be backed, and LW can probably influence the video creators in a useful/helpful way.