In practice there is no such thing as a perfectly rigid rod. All rods compress to some extent, so they are like extremely stiff springs, but they are such stiff springs that you can ignore the compression. Examples and exercises in physics are usually idealised (eg they might assume a perfectly round wheel or or a perfectly flat surface), so a rod might be considered to be perfectly incompressible for that reason.
If you bear in mind that work is force multiplied by distance, then when you compress a spring the distance through which it is compressed will usually be significant, but for a perfectly incompressible rod the distance will be zero, hence the work done when you apply a force to try to compress the rod is zero.
The explanation that internal forces, being passed through touch, do not add work, is simply misleading. At a microscopic level there is no such thing as touch. Forces are passed through solids as a consequence of electromagnetic forces which arise when the particles comprising the solid are perturbed from their equilibrium positions, either by compression or extension. The degree of compression or extension depends upon the physical properties of the material. In a stiff solid, a force can be propagated with very little compression, so little work is done. In a more flexible material, the particles in the solid move further from their equilibrium positions when a force is propagated through it, so the material compresses or extends more, and more work is done.