Antennas work by accelerating electrons. Accelerating electrons emit photons.
Since the EM field is a gauge field, phase matters. Thus an "antenna" a full wavelength long has an average phase of 0º and doesn't emit many photons. For every electron moving one way, there's another going the opposite way. But an antenna that is ½ wavelength long has all the current adding part of the time. So 180º phase radiates well.
There is a principle called reciprocity which states that antennas which emit well also receive well. I'm not fully clear on why.
Something called image current also exists. An image current is caused in a conductor when electrons (or other charges) move near it. The negative charge of the electron draws a corresponding positive charge which exactly matches the electron's movement.
This image current can be used to adjust the electrical length of an antenna. For example a quarter wave dipole creates another quarter wave in the ground plane like a mirror. This adds up to a half wave which makes a good antenna.
Finally, an antenna attaches to a transmission line. It acts as an impedance transformer between the transmission line and free space. Transmission lines can be any impedance, but 50Ω is typical. Air has an impedance of 377Ω. So the antenna typically needs to have an input impedance of 50Ω. Mismatching impedance causes energy to reflect back to the power supply, causing problems.
Understanding these principles, let's look at the Biquad:
The antenna is a pair of squares of wire above a ground plane. Each side of the square is ¼ λ. The squares are arranged like a bow tie. Typically they are fed from the center of the tie with an unbalanced coax cable. The center conductor is attached to ½ of the center, and the ground is attached to the other, the post, and the ground plane.
Let's look at the current distribution. The current is fed as a sine wave. Let the feed point be 0º. So at t=0, there is no current at the feed point. The current is maximum at the end of the first leg, and zero again at the end of the bow. It then goes negative and back to 0 at the ground feed. This happens in both squares of the bow tie shape.
Since the square legs are orthogonal. Thus only the legs in the same direction add phases to signals at one polarity. (The overall wave will combine phases from the two directions, but we will only look at one.)
A point far from the antenna will see the positive current (which is made of accelerating electrons) and the exact opposite current. But these opposing currents are 90º out of phase. In addition it will see an opposing image current off the ground plane, again 90º (nominally anyway) out of phase. This adds up to two identical ½ dipoles in phase for each of the two squares. That's a total of 4 ½ wave dipoles.
But that's only the signal polarized in one direction. There's and identical antenna set with orthogonal polarization. So 8 dipoles worth of power are directed outward. That's about 8-9 dB.
The input impedance depends on the geometry, including the feed points and the distance between the ground plane vis a vis the wire diameter. (Remember how the ground plane spacing was nominal? It needs to be adjusted to match impedance.)
This should help you understand what's happening. You can model the actual fields with the right software and or analytical calculus.
When building this, it helps a lot to have a Vector Network Analyzer. Getting everything just right is hard, but you can tweak it on the VNA.
Finally, there are legal issues. The FCC power mask is set as output power from the antenna, so adding directivity to the antenna means you need to reduce the output power correspondingly. While I doubt the FCC goes around checking WiFi specifically, if there's ever a problem with interference, bad things could happen with lawyers and judges.
Good luck!
