Five Cubes in Dodecahedron

A cube has twelve edges, and the cubes you're talking about have one edge along each of the twelve faces of the dodecahedron, on which the edge is a diagonal.

The faces of the dodecahedron are pentagons, with 5 diagonals.

If you start with a given diagonal of a face of the dodecahedron, it uniquely determines one of the five inscribed cubes. You can show what it must look like near a vertex. (See the figure below.)

Once you have this, make a sketch of the dodecahedron near a face of the cube. The dodecahedron over each cube face has a "tent" (double-y, bent-H, what have you) shape:

You should be able to make a convincing argument, based on symmetry, that what you get is in fact a cube, and it does "close up" into a coherent shape when you follow the rules for drawing diagonals.

Practice making sketches with four faces of the dodecahedron showing, and use a different color for the diagonals which form the edges of the cube.