Note that it is not a mesh that is used but it is a sheet of perforated metal. That is easier to clean and more conductive. If the dimensions of a hole are all less than about one tenth of a wavelength then microwave radiation cannot radiate through the hole. You can see inside through the perforations because light has a much shorter wavelength than microwaves. The outside of the perforated sheet is painted black to make it easier for you to see inside without problems from external lighting.
The 2.45GHz microwave radiation used has a wavelength of 12 cm so you might ask why they do not use 12 mm holes? The answer is that, from a distance, you can see through many small holes better than through a few large ones.