First, you have to notice that there is no "speed of light", there is more an effect that is electromagnetic, and its propagation in the fabric of the world, which is " space-time ". The fabric of our world is not space only, and is not time only, it's both

So what travels at this speed that we call "c"? Well, it's causality. What we see traveling at the maximum speed is the propagation inside the fabric of our universe of the forces (electromagnetic, gravity, atomic weak and strong forces, etc)

And this travel is wave-like, we know that because of light. So it's not that light is a wave, it's that light is electromagnetic, and the propagation of causality is a wave

And because of that, it was postulated by Einstein as a consequence of the theory, that there should exist waves of gravity. Einstein even believed that there won't be any way to measure them, but eventually people found a way, and also found a possible cause of natural event that could generate some, which is the collision of 2 black holes