It's not about neural network architecture. CNNs are taught by presenting them overlapping pieces of image. To speed things up and keep things orgsnized this is not done sequentially but in parralel making multiple neurons share weights but this is just a trick.
So what makes this result possible is not the architecture of NN in CNN but rather architecture of C. That allows us to get multiple samples from single image. The rest is just that actual content of the image is easier to learn then the noise.
I think it's both C and NN. Don't forget each new C layer groups information from previous layers; using just a single C layer won't do you much good. It might not reflect brain much but it kinda resembles what retina/visual cortex neurons do; CNNs were actually inspired by visual field maps found in visual cortex and somebody had the idea that C is the most similar CV operation we have, and put them together. To everyone's surprise it worked nicely.
It's probably just very rough "resemblance" :D It is said CNNs were "inspired" by visual field maps; I am 100% sure we know very little about how that part of brain works and maybe somebody just took a look at main/thickest connections between neurons there and tried to assemble them in a NN to see if it helps.
So what makes this result possible is not the architecture of NN in CNN but rather architecture of C. That allows us to get multiple samples from single image. The rest is just that actual content of the image is easier to learn then the noise.
Brain is almost nothing like CNN.