What's the role of Ground Plane in high frequency boards?
Specifically, why is that via indicated by red-arrow on the layout, at the top left?

Because that via is grounding one side of a bypass capacitor; we don't know the level of current coming through that capacitor, but the human designer did not want that current to flow on the edge of the layout, where "slit antennas" are a bad idea and will radiate. Radiation inside boxes will cause coupling of circuits located far apart, and upset the gain flatness, the phase consistency, and possibly produce oscillation.

View every via as 1 nanoHenry of inductance. At 10GHz, that is j63 ohms of impedance. Wider holes give a lower impedance. In this example of yours, NOTHING can be absolutely clean or quiet, because of all the bits of metal. 

As I was taught in past decades, "ground is just a cruel joke".

On the top right, the reasons are the same. Particularly those large black blobs with 4 thick pins out the 4 sides.

Don't view RF as magic. Just continue to expand your awareness of all the pieces of metal, all the vias, all the opportunities for coupling. Sometimes even the dielectrics are used for signal paths.