I watched less than half of this video, but I think I was able to figure out how they do this "hyper-lapse" technique.

The first thing I noticed is that the finished results show less than the original speeded up footage. As in they look more "zoomed in".

What I think they are doing is taking each frame and cropping it in a way that removes the shaking and makes it look smooth.
If the original shakes to the right, they cut off the right edge. When it shakes up, they cut off the top of the frame. ect.
This creates a more fluid and smooth looking video, but some of each frame is lost.