How does this circular polarized antenna work?
Let's take this on from the cable:


The discrete component is probably just something to help with matching the microstrip wave guide on the PCB to the cable impedance. Since that way of soldering the cable to the line certainly isn't the most well-defined one in terms of reliable impedance, that match doesn't need to be overly great.
The ring-like structure is a balun, which converts between the unbalanced signal from the coax and the balanced signal going into the branches of that "J"
That J is has different branch lengths, meaning that the right circular patch gets the same signal with an advanced phase compared to what the left patch sees.
These patches are a bit unusual in their size relative to wavelength, which is why I'm surprised this antenna actually emits nicely normally to the ground plane. Clearly, their phase and shape effects the polarization of the emitted wave, but I fail to see how the circularization works.


EDIT

now after you've added the aluminium disc:

That's the actual antenna; the small patches just act as feeds for the holes in the larger aluminium disk.

So, this is just a helical antenna!

Imagine that you take a helical antenna, and cut it along a plane: That's exactly the plane that the disc lies in. The currents you induce in that disk are just emulating the current distribution that would cause a field identical to that of the helical antenna.