Why the RMS of a PSD curve is the root of the area below
I think this is simply an aspect of Parseval's Theorem (e.g. click me)

It simply says: sum of squares in the time time domain equals sum of squares in the frequency domain. Substitute "sum" for "integral" if using the analog domain. In other words: total energy in the time domain equals total energy in the frequency domain. 

This can easily reproduce your formula, $g_{RMS}$ represents the  time domain energy and the integral on the right represents the energy in the frequency domain. The exact scaling depends on the details like length of the signal, periodicity, sample rate, etc.