The platonic graphs arise from five solids: tetrahedron, octahedron, cube, dodecahedron, and icosahedron, where the vertices of the graph are the same as the vertices of the solid, and the edges of the graph are the same as the edges of the solid. Of course, these graphs can be morphed in multiple ways as long as each vertex is connected appropriately.
So let's look at the graphs of the tetrahedron, octahedron, and cube:
You should be able to visualize how the graph of the tetrahedron and cube has resulted. The others follow in the same way.
Regularity of the Platonic Graphs
Below is a chart describing the regularity of the 5 platonic solids:
Verify for yourself that this is true for the tetrahedron, octahedron, and cube graphs in the earlier figure.