First of all, I'm not an expert, but that can be an advantage in trying to explain the equations in lay terms...
Maxwell's equations are these, in differential form:
- $$ \nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$$
- $$ \nabla \cdot \mathbf{B} = 0 $$
- $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $$
- $$ \nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) $$
Eq. 1 means that an electric charge creates a proportional electric field .
Eq. 2 means that there is no magnetic charges (monopoles).
Eq. 3 means that a variation in the magnetic field creates an electric field.
Eq. 4 means that a variation in the electric field plus currents (moving chages) creates a magnetic field.
Now, with that in mind think of what all this means: an electric charge creates an electric field, the charge moves, the electric field changes, that change creates a mangnetic field... All the known electric and magnetic phenomena.
But, hidden in these equations there is another interesting possibility. Without the need of any electric charge, a sinusoidal electric field could create a (cosinusoidal?) magnetic field, that in turn will create another electric field, and so on. That is, a standing electro-magnetic wave, without the need of any charge at all!
Those are what they call solutions to the Maxwell's equations.
Now, you want to create a radio transmitter. The simplest form is just an antenna, that is a long piece of wire, in which you inject an electric current that oscillates at the same frequency you want to transmit (from a few kHz to hundreds of MHz). The variable electric field in the wire will create an electro-magnetic wave.
The funny part is to build the receiver. For that you will need another antenna. There, the coming electro-magnetic wave will induce a current (fractions of a mA), that you will amplify with an electronic circuit (a valve or transistor will do) and send to whatever device you use to generate the final output.
