The following grasshopper definition is meant to explain the concept of transitivity with the pentagonal dodecahedron (mind that its upper face is missing). Below the video, a description of what is going on.

The first axis to emerge, in red, is a 5-fold axis of rotational symmetry. If we rotate the dodecahedron around this axis, the vertices, edges and faces will move to a position that is indistinguishable from the initial, provided that the angle of rotation is 1/5 of the complete turn, that is, 72º, 72º more and so on until we get to 360º.

The second axis to appear is a 3-fold axis of rotational symmetry. Again, we rotate the dodecahedron around this axis, and it moves into a position indistinguishable from the first at consecutives 1/3 of a complete turn: 120º, 120º more, until a third gives us the complete turn.

With each of the 2-fold axis of rotational symmetry, there are only twp positions in which the dodecahedron moves into a new position that is indistinguishable from the initial: at half a turn (that is, at 180º) and at 360º.

For the regular dodecahedron, the number of 5-fold axis of rotational symmetry is half the number of faces (6); the 3-fold axis is half the number of vertices (10); and the 2-fold axis, half the number of edges (15). That is, 31 axis of rotational symmetry in total: six 5-fold, 10 3-fold and 15 2-fold.

With the necessary adaptations, the same happens with any regular polyhedron, because they are all transitive in their faces, edges and vertices (they are flag-transitive, but I will not address the concept of flags here).