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I have heard that in an antenna the electrons move back and forth and create an electric field which varies with time. This varying electric field in turn, creates a varying magnetic field. These two varying fields together create an electromagnetic wave.
How do these waves travel out from the antenna?
If the electrons are moving back and forth vertically, wouldn't there only be an electromagnetic wave near the antenna? What makes them spread out or expand?
Antenna is a device, consisting of horizontal stick with aplied current on both ends. The current is alternating and varies harmonically (for simplicity) $I \sim I_0 e^{i\omega t}$.
From Ampere's law we know that current will generate harmonically varying magnetic field $B\sim B_0 e^{i\omega t}$, which from Faraday's law will generate alternating electric field $\nabla \times B = \frac{1}{c^2}\frac{\partial E}{\partial t}$, where $E\sim E_0 e^{i\omega t}$.
From Maxwell's equations:
$$
\nabla \times ( \nabla \times E) + \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = 0
$$
This is a wave equation that tells you how your EM waves propagate. In case when the size of antenna is much larger than the distance between observer and antenna, you will have your wave propagates perpendicular to the antenna in horizontal direction. However, when you will go further and further from antenna, it could be viewed as a point source and your wave will be radially symmetric.
In a similar on hold question of yours, I replied to how what is called the classical electromagnetic wave propagates from an antenna. It is a matter of measurements and observations that it does, and the boundary value solutions of the classical Maxwell's equations fit the observations and are successful in predicting any new electromagnetic radiation setup. So the answer classically is "because that is what is measured" .
So classically the how ends there . We have progressed though and we know that the world is quantum mechanical and at base it consists of elementary particles and the current successful model is the standard model. We can then, in the quantum mechanical frame, go the lower level of "how" an antenna radiates electromagnetic waves, by noting that we have found experimentally that a light wave consists of photons, elementary particles in the standard model.
They can be released one at a time, and are small dots on the display, but as their number increases they build up the classical interference of light of the given wavelength.
So we can go one step further on the how: The electrons moving in the conductor of the antenna are radiating individual photons, which in concert with innumerable individual photons from other electrons emerge with velocity c in the space surrounding the antenna. They build up the classical electromagnetic wave according to the mathematics of quantum field theory.
Now why light consists of coherent motion of zillions of photons is answered by "because that is what has been observed and measured".
Because I think you don't have the knowledge others on this site believe you have, I think my answer Will help you the most, but it's a simplification in order to make you intuitively understand what is happening.
The electrical fields (changes of value of electric field, before people start downvoting) are constantly being "created" by electrons even if they are not moving. Those fields are constantly flying away from the source. But because the 3 spacial dimensions of our universe, they are getting smaller in value by distance squared. This is why the electrical field is stronger, the nearer you are to the electron and gets considerably weaker when you move away from it.
When you start moving the electron, you Will see that change in electrical field only after that change in electrical field Will reach you with the speed of light.
Now, when we have alternating current in the antenna, that is generating a changing magnetic field around that current, that is then flying away from it at the speed of light. As we know, changing magnetic fields generate a changing electric field and that generates a changing magnetic field again....
The energy carried by a changing electric and magnetic field is quantatized by what we call a photon. But things get much more complicated from here on.
A pure carrier wave signal (no audio information / modulation) is basically just a high-frequency AC sinewave generated by the transmitter.
A conductor with a potential at the ends creates an electric field along the wire, and a magnetic field revolving around it at right angles.
At RF frequencies above (30 KHz AC) these fields start to radiate outwards as electro-magnetism (EM waves).
The electrical field E is parallel to the antenna length, while the magnetic field H, still in phase with the E field, is at 90 degrees to antenna (follows H field direction from wire).
When the transmitter reverses polarity, both E and H(M) fields also reverse polarity.
This polarity reversal repeats at the frequency generated by the transmitter.
In the book "Modern Electronic Communication", extracted text describes the basics (long article abbreviated for post):
Electromagnetic Waves: All electric circuits contain E & M fields. In circuits this energy is usually returned to the circuit when the field collapses.
If the field does not fully return its energy to the circuit, the wave is partially radiated, or set free, from the circuit.
An efficient, resonant antenna is designed so as not to allow EM energy to collapse back into circuit.
This is due to current flowing in opposite direction (nul).
Thus this reverse-polarity (negative) half-cycle of waveform is also transmitted.
The E and H fields interact with each other, much like a generator/motor (electricity<>magnetism).
The wave is said to be traverse since the oscillations are perpendicular (at 90 degrees) to the direction of propagation (wave travel).
Antennas have polarization (vertical or horizontal) which is determined by the direction of its E field component.
As a rule of thumb, transmission and receiving antennas should be same polarization for maximum efficiency.
For a graphical animation of EM radiation, navigate to Physics Stack Exchange>[Electromagnetic-Radiation]>@asmeurer (answer).
