The first step to successfully seeing the partial cubes turn with you lies in your ability to perceive them as solid, six-sided objects. This perception has a lot to do with being raised in a society that recognizes cubes as a common shape. Your brain is used to seeing cubes, so even though this partial cube only has three sides, it assumes the cube is solid.
As you move past the assembly, your view of the corners changes in a way that would not make any sense if the corners were stationary cubes. Your eye-brain system is accustomed to seeing things that are near you move faster than things that are farther away. When you’re riding in a car, for example, nearby objects seem to whiz by, whereas distant objects seem to follow you at a slower pace. Because you perceive this inside corner to be the outside of a solid cube, your brain “sees” the corner farthest from you as being the closest. To maintain this misconception, your brain perceives a rapid rotation of the cube as your angle to the corner changes (click to enlarge the diagram below).
The diagram above shows how this illusion works. In the real situation, as your eye moves to the right, it sees more of side A. In order to see more of side A of the imagined corner, the perceived cube must be seen to rotate as you move.