They mentioned that 3D space means you can pick any point in the space, and draw 3 axes out from that point, all of whom are perpendicular to the other 2. Easy enough to understand. But a 3D representation drawn on a 2D space (at 0:09 of the video) requires that some angles are mutated (the ones between red and yellow, for example). This is because in 2D space you cannot represent "the third axes perpendicular to the other two". However, our brains see that shape as 3D, probably because we live in a 3D world so our brain can "see" the 3 dimensional-ness of it.

Same goes for the 4D simulation. It is actually 4D, just that in a 3D space (2D, actually, since your computer screen is 2D) you can't draw all angles perpendicular to each other. Take 2:23 on the video, I think we can agree that green makes a cube, yellow too, one cube with the top shaded purple, one with the top shaded blue, one facing us with "yellow, red, green, red" vertices on top, and one at the back.

If a 2D square is made up of four 1D lines, and a 3D cube is made up of six 2D squares, what is a shape made up of six 3D cubes? 4D, of course, says my mathematical brain.