Ask a Physicist
Why does combining the principle of relativity with the invariance of the
speed of light lead one to conclude that no medium is required for light to
First, why do we get light propagating at all?
James Clerk Maxwell's equations for the electric and magnetic fields imply special relativity, though it wasn't obvious at the time (Einstein wasn't born). They led him to predict that there could be waves in the fields, and that these waves were in fact light. Up to then, people thought that electricity, magnetism, and light were unrelated.
Qualitatively, the equations mean this: Electric fields point out from positive charges and into negative charges, and magnetic fields curl around currents (moving charges). Those are the main things you notice in a lab, which is full of matter, and historically they were the first to be discovered by playing with bits of wire and amber and so on. If that was the whole story, you wouldn't see much in a vacuum because there aren't any charges or currents.
But electric fields also curl around magnetic fields that are changing with time, and magnetic fields curl around electric fields that are changing with time, never mind any charges or currents. Those effects were harder to detect, especially the latter which is very weak (but you might know the former as "induction"). Maxwell put them into his equations and noticed that if you set up electric and magnetic fields curling around each other and changing with time just right, they can make waves that will propagate off and sustain themselves even in vacuum where there aren't any charges or currents. And those waves have to travel at exactly the speed of light, which was too suspicious to be a coincidence.
Because you get this mutual curling even in the absence of matter, it doesn't rely on a medium to exist and can be perfectly happy in vacuum. In fact, light and other electromagnetic waves slow down when going through matter (unless it's opaque and they can't go at all).