The antenna is a half-wave low impedance transmission line.
It is fed at or near the edge with a mainly voltage drive across the capacitive open circuit. The centre of the patch is a voltage node so looks like earth. So you have the first 1/8 wavelength looking like a capacitor with a voltage excitation, and the next 1/8 wavelength looking like a an inductor to ground (across the capacitor).
This is a parallel resonant circuit. If it was fed at or near the centre it would be a series equivalent circuit. If you measure it on a VNA you will see the resonant loop on opposite sides of the Smith chart for the two cases. It doesn't matter where the resistor is provided it give the right losses by having the correct equivalent value, which are mainly series losses for the conductor losses and shunt losses for the dielectric losses.
The second half of the patch has the opposite voltages and keeps the voltage node in the centre. You could have a shorting wall across the centre, but that would only give one radiating slot and give some gain reduction.