To get to the heart of the matter, you need to understand two things:
- Maxwell's equations
- Fourier analysis
Maxwell's equations tell us the relationship between electricity and magnetism. In particular, they tell us that a change in an electric field causes a magnetic field. Similarly, they tell us that a change in a magnetic field causes a change in an electric field. Maxwell's equations are also differential equations that we can solve (in some circumstances). If an electric field is sinusoidal, then the associated magnetic field is sinusoidal as well, and timed in such a way that they reinforce each other. This means that energy gets transferred between the electric field and magnetic field and back again. This is what we call an electromagnetic field. It is a self-perpetuating wave.
Now, the other side of the story is that a stationary charge will produce a static field. But if the charge moves towards you, the field strength will increase. In particular, this means that a magnetic field is created as well. The movement of the charge does not have to be sinusoidal. Basically, we can use Fourier analysis to add up any changing electric field into a sum of sine waves. And the associated magnetic field will be a sum of cosine waves in the way we expect. The point is, every motion of an electric charge will create some waves.
An antenna works in reverse. There are basically two types. Those that absorb magnetic energy and those that absorb electric energy.
A pure wave will have an associated wavelength, and if an antenna is exactly half that wavelength, then the difference between the field at one end and the field at the other end will be maximal (for the given wave). When the wave passes over the antenna, it will effectively set up a voltage difference between the "high point" and the "low point" of the wave.
A magnetic antenna works in a similar way, but it captures magnetic energy. And it's harder to visualize. But basically, a magnetic antenna is a fancy tuned inductor.