Here's an attempt at making sense of that intuition. Let's compute the volume of a pyramid that's inside a cube. Imagine that you're inside a cube room (this will make it easier to identify the faces). You are sitting in a corner.

From your corner draw lines to the four ceiling corners -- a cube diagonal, two face diagonals and a cube edge. This is the pyramid. Now draw lines to the corners of one of the walls opposite you -- it's the same pyramid, but on its side. And lastly draw lines to the other wall opposite you -- again the same pyramid. You have now covered the entire cube with three identical pyramids, so the volume of the pyramid is a third of the volume of the cube. The interesting part is that the bases of these pyramids are on the three dimensions, giving the intuition that the /3 is due to the dimension. This proof might even generalize in higher dimensions.

Edit: I think this is also the gist of the top answer in TFA, but it's weirdly formulated imo.