Exactly the same happens as when light reflects off a metal surface. In both cases you have an electron gas that interacts with the light. In the case of a metal it's a dense (almost) free electron gas, and in the ionosphere you have a very dilute electron gas formed by ionisation of air molecules.
The incoming electromagnetic wave causes the electrons to oscillate, and as the electrons oscillate they emit EM radiation. If the forward direction the induced radiation emitted by the electrons interferes destructively with the incoming wave, and in the reverse direction induced radiation emitted by the electrons interferes constructively with the incoming wave. The result is that the wave is reflected.
The dense electron gas in a metal interacts with the incoming EM radiation so strongly that even a micron thick layer of metal is effectively perfectly reflecting i.e. reflects 100% of the incoming light. Because the electrons are so dilute in the ionosphere the reflection is far less efficient, though of course the ionosphere is a lot think than a micron and anyway you don't need perfect relection for radio transmission.