What is the phemonemon behind electromagnetic waves being able to propagate in the vacuum?

Many concepts in modern physics boil down to looking at something that worked in classical physics, and realizing that apparently crucial features are not really as important as they appeared at first glance.

Your description of the propagation of electromagnetic fields is spot on. While it's very counterintuitive for us to imagine a wave propagating without a medium, in fact it turns out that there is no need for particles to move for a wave to exist. The crucial feature of a wave (in our current understanding) is that energy is transported from one region of spacetime to another. Maxwell's equations allow propagating disturbances in the electromagnetic field to carry energy.

The trick is that these disturbances do not imply the movement of any particle and there is no rest frame for a medium in which the waves propagate. But it turns out this is just extra baggage that our common experience causes us to expect, but which is not really necessary. If you look at the equations (Maxwell's equations) defining the laws of motion for the electric and magnetic fields, you can show that the electric and magnetic fields obey a wave equation (ie -- the same equation obeyed by small ripples on a pond, or waves on a string). This equation has exactly the same form in any reference frame. The modern point of view is to take these governing equations as the definition of a wave -- even though this is an abstraction from our common sense intuition that waves should involve a disturbance in a medium.