This is s tough one....I don't even think the guy who invented it understands it!
But here's a thoroughly inadequate explanation, for whatever it's worth.
A dielectric resonator is functionally similar to a piezoelectric crystal, except the resonant frequency is determined not by physical vibration on the large scale, but rather on the MOLECULAR scale. The main characteristic of a dielectric is that it slows down radio propagation, as compared to air or in a vacuum. The most IMPORTANT phenomenon is what happens at the BOUNDARY between the dielectric and the air...there is a sudden change of velocity which results in a total reflection at the surface back INTO the chunk of dielectric material. Because of the slow velocity of propagation, the dielectric slug can be made very small, compared to a similar "cavity resonator"
R.F. input into and out of the resonator is generally by means of capacitive coupling, just as in a piezo crystal.
Hope this helps a little. There are entire PhD theses on how these things work, if you're really interested.