I would just explain that a photon has zero mass, but nonzero energy and momentum and finite velocity. Intuitively (meaning: in a Newtonian universe), one would think that a particle with zero mass would have to have infinite velocity to have nonzero energy and momentum, right? And yet, its velocity is finite. That demonstrates that the universe is not Newtonian. It also sets up an intuitive connection (which is what we're looking for here, right?) between the speed of light and one's intuitive sense of an infinite velocity.

Again, of course, this brings us to the question of why the universe is this way: why it is Einsteinian rather than Newtonian. But that question really belongs to the realm of metaphysics, not that of physics.