Why do we see shadows in the following black-and-white optical illusion? How does neuroscience explain the following optical illusion? Look at tiles. You will see spots of shadow near all corners. The shadow will be missing from some corners very near the center of your attention.

So, the brain has ganglia that detect horizontal and vertical edges as well as more complex edges. They work best with high contrast edges. That's all I know right now that might be relevant. Thanks in advance.
Looks like these are variations on the Hermann grid illusion, wiki page here: https://en.wikipedia.org/wiki/Grid_illusion

As the wiki page notes, the first-pass explanation of this was lateral inhibition, but this is not really 100% satisfactory. 
There's a nice quick walk-through of the old explanation and the problems with it here: http://web.mit.edu/bcs/schillerlab/research/A-Vision/A15-2.htm 
A journal paper version of this stuff from the same lab is:
Schiller, P. H., &amp; Carvey, C. E. (2005). The Hermann grid illusion revisited. Perception, 34(11), 1375-1397.