I found the visual justification for partitioning a cube into three pyramids to be a bit confusing. For me, an easier way to think of it is that there are three coordinates (x,y,z) and each pyramid represents a region where a particular coordinate is largest. e.g. One such pyramid is {(x,y,z) | x = max{x,y,z}}.
Pick a corner of the cube. There are three faces that go through the opposite corner, right? The pyramids are formed by the first corner and those three faces. (They're obviously equal.)
Now imagine looking at the inside of the cube through that first corner. There is no angle where you don't see one of the three faces in the background! So the decomposition is complete.
My preference is to take the point in the center of the cube and make pyramids to each of the 6 faces. There are twice as many pyramids but they are half as high.
