Recently, there has been an increased interest in the problem of wave transmission through sub-wavelength apertures, following successful experimental demonstration by several groups for enhancing optical power transmission through nano-scale holes in metallic screens due to properly designed periodic corrugation. Oliner, Jackson, and their co-workers explained and justified this phenomenon as the result of the excitation of the leaky waves supported by the corrugated screen. Here we discuss in detail the mechanism and analysis for another setup we have recently proposed, in which metamaterial layers with special parameters may be utilized as covers over a single sub-wavelength aperture in a perfectly electric conducting (PEC) flat screen in order to increase the wave transmission through this aperture, and we provide a detailed physical insights and analytical explanation for this aperture setup that may lead to similar, potentially even more pronounced effects when the proper metamaterial layers are used in the entrance and the exit face of the hole in the flat PEC screen with no corrugation. Some numerical results confirming this theory are presented and discussed. We also investigate the sensitivity of the transmission enhancement to the geometrical and electromagnetic parameters of this structure.