The patch antenna array voodoo: four patches, each fed at two points at 90 degrees, with a 90 degree (1/4 wavelength) phase shift delay made out of PCB track microstrip transmission line thus producing circular polarisation.

To maintain the right impedances, there are a bunch of 1/4 wavelength transmission line transformers, i.e., 1/4 wavelength lengths of different width PCB tracks on top of the ground plane making various different impedance microstrip transmission lines: you could use several pieces of 1/4 wavelength different impedance coax lengths if you wanted to, a transmission line is a transmission line, but it's easy and cheap to fabricate on a PCB together with the antennas. The transmissions lines are not designed to radiate, just the square patches are meant to do that!

By using a 1/4 w/l transformer, you can change the impedance in a reasonably loss free way.

For example, say you have two 50 ohm antennas you want to feed from a single 50 ohm source, if you just parallel them up you end up with 25 ohms. But by placing a 70.7 ohm 1/4 w/l transmission line in front of one 50 ohm antenna, it will look like 100 ohm. Do that twice, once for each antenna, and parallel the fwo 100 ohm presentations together, you get a single 50 ohm presentation. Bingo!

The 70.7 ohms comes from Zt = sqrt(Z1 * Z2) where Z1 and Z2 are the impedances you want to transfer from and to, and Zt is the impedance of the 1/4 w/l transmission line required. So in the example, Z1=50, Z2=100 so Z2=sqrt(5000) = 70.7 ohms.

Now, the velocity of the wave on PCB or any substrate isn't as fast as in free space. The velocity is c/sqrt(Er), so say for FR4 with Er approx 4, the velocity factor is about 0.5, so the length of the 1/4 w/l microstrip lengths would be about half that had they been made in free space.

To figure out the impedances of the transmission lines, predominently it's calculated from the width of the track, height above the ground plane and the permittivity (Er). 50 ohms is about 3mm on 1.6mm FR4, there are on line calculators for this.

FR4 might be OK to L band which this antenna looks to be judging from the size, but it becomes quite lossy much higher in frequency.

Edit: Why the foam? This is a physical mechanical separator between the patch antennas and the plastic enclosure. Because the foam's mostly air (Er=1), it won't detune the patch antennas and transmission lines, which a very close proximity plastic enclosure would do.