First. Eletric and magnetic field is dependent of the inertia frame. Only when we see this two fields together, that we have an physical object living in space-time, being independent of the inertial frame. Actually, when we do this we learn what is the truly symmetries of our space-time. Einstein is the one that discover that ;).
Radio waves are compound by electric and magnetic field. One create the another through space and time. See Maxwell's equations that describes this process.
Let's go to you question:
Behind the electromagnet (EM) field we have photons. But this is in some way a quantum relationship. We can say that photons and EM field is two faces of the same thing, as a generalization of wave-particle duality. In some cases we may have a bunch of photons propagating through space building an EM wave. In other cases we may have a state builded by a lot of superpositions of photons created from the vacuum propagating through space and annihilated to the vacuum, that constructively build an stationary EM field state. Actually the vacuum itself is build in that way. This process is more clear when you understand Feynman diagrams, that is a perturbative application of Path integral formulation of Quantum Mechanics.
This is quantum mechanics, so don't think that this particles are actually there. Only if you put an delicate detector you would measure they. But in quantum mechanics before the measure we have only potentialities of possible outcomes and they probabilities. And after the measurement you destroy the field state or vacuum state, in case of local measurements.