The explanation is as follows.
We know that light doesn't need a medium through which to travel because the speed of light is experimentally constant: independent of the movement of the source or detector or the direction in which it travels.
Light contrasts with sound, which travels through the air (or some other material medium). If you're stationary with respect to the air, then the speed of sound is the same in all directions. But if you're moving with respect to the air, the speed of sound will be the same in all directions relative to the air—which means that sound coming up in front of you will seem faster and sound catching up to you from behind will seem slower.
If light were a disturbance in a medium, it would exhibit the same behavior. But light never does—its speed is the same under all circumstances. So it does you no good to postulate an aether. You can still do it, but it makes the theory more complicated than necessary. The only reason to postulate an aether is that you're uncomfortable with the idea of waves not needing a medium. But our modern understanding of quantum mechanics is that all kinds of particles have a wavelike nature, so, if you accept that matter can travel through empty space, you should have no problem accepting the same for light.
So in short, light do not need a medium to propagate. The mechanism involves photon propagation.
Hope it answers your question.
All the best!