The diameter-to-the-fourth relationship is easy to understand. If you are looking a hole in a metal sheet, the size of that hole represents the source of the radiation coming through from the other side. If you have two such sources far apart, then you get twice as much power. But if they are close together...closer than the wavelength of the radiation...then they represent two coherent sources whose amplitudes add. So you get FOUR times the power. Diameter-to-the-fourth. (Because it doesn't matter if the two holes are merged into one.)
I don't think this is helpful for explaining the mechanism of the Faraday cage. I think the Faraday cage becomes highly effective when the cage dimension is anywhere close to the wavelength...in other words, I think the attenuation is probably much more than 10,000 for a cage spacing of lambda-by-10. (More precisely, I think if the attenuation is x at lambda-by-y, it is much more than 10000x at lambda-by-10y.)
I'm basing this gut feeling on the analysis I did of the parallel-wire filter here:Particles, waves and parallel wire filters. Transmission formula?
The key to this analysis is to look at the reflection/transmission properties of a conducting sheet. Naturally, if the resistance is zero, you get total reflection. The funny thing turns out to be that if you add resistance, there are no solutions whereby
(incoming power) = (transmitted power) + (reflected power)
because all solutions must include some power loss in the resistance of the sheet. So for the faraday cage, assuming no resistance, once the wires are close enough together, where are the losses? So it becomes very difficult for the filter to do anything other than total reflection.