How does amount of free electrons in an material relate to it's ability to reflect? From this Wikipedia article on critical frequency, it turns out that there is a limiting frequency at or below which the radio waves are reflected back to the earth. The reason for such a frequency is said to be electron limitation and said by wiki " The inadequacy of the existing number of free electrons to support reflections at higher frequency" but how do electrons in the ionosphere reflect EM waves?
 A: Roughly speaking, when an EM wave hits an free electron, the electron starts to oscillate in response to the electric and magnetic fields in the wave.  Since the electron is now oscillating, it creates its own electromagnetic waves at the same frequency as the incident wave.  In particular, there will be some waves travelling back towards the source of the original EM wave.  What's more, the interference between the original wave and the scattered wave means that the total amplitude of the wave traveling past the electron (i.e., the transmitted wave) will be reduced.
In this picture, it's relatively obvious why the density of the electrons matters for the amount of reflection vs. transmission that occurs.  If there are relatively few free electrons, then their scattered waves are weak and the incident wave and transmitted wave will be basically the same amplitude.  On the other hand, if there are a lot of electrons around, their combined scattered waves will be sufficiently large to completely destructively interfere with the incident wave in the region past the electrons, and create a new reflected wave with basically the same amplitude as the incident wave.
A: Electrons in ionosphere can be described as charged particles in a harmonic potential (due to positive ions). That is, they experience elastic force $-m\omega^2 x$ in addition to force from the external EM wave; here $\omega$ is called the plasma frequency and is a natural frequency of oscillation in such plasma. It depends on the concentration of the electrons (in the model there are also positive ions in the plasma so the medium is neutral).
It turns out from analysis of equations of motion of such a model that if the frequency of the external EM wave $\Omega$ is much lower than $\omega$, then electrons oscillate in sync with the force due to the external EM wave and thus produce an EM wave of their own, a so called secondary wave. We we add the waves together, net EM field looks like very weak wave passes through, and a wave of similar strength to the primary one gets reflected back. So we have near total reflection. For low frequency radiation, ionosphere behaves as a shiny polished layer of metal, a mirror.
When the EM wave frequency gets close to $\omega$, the plasma begin to absorb the EM wave energy a lot, it gets hotter and only very weak EM wave gets reflected and transmitted.
But if $\Omega$ is much higher than $\omega$, then electrons do not manage to keep up with the external wave, so their oscillations get only very small amplitude and they produce only very weak secondary EM wave which can be neglected. Then the result is the original EM waves passes through the plasma layer almost unchanged.
