Like most people sharing this around today, the Neural Correlate Society's "Illusion of the Year" blew my mind a bit the first few times I watched it. But, even then, I got the sense that this is probably pretty simple and the illusion probably has to do with the actual real geometry of the shape, which doesn't appear to resolve to a proper circle and also doesn't seem to have the shadows we'd expect with a proper circle.
Googling around, I found a brief paper explaining the "ambiguous cylinders" from the illusion's creator, Kokichi Sugihara of the Meiji Institute for Advanced Study of Mathematical Sciences in Tokyo. Indeed, the explanation is that the shapes in reality are just perfectly in between cylinders and squares, geometries that the eye naturally wants to resolve into one or the other. If you pause the video at :15 seconds, you can see that the tops of the shapes are not actually flat.
"Although we know in our logical part of our brains that we are looking at the same object directly and through the mirror, our brains usually do not correct this kind of contradictory perception," Sugihara writes. "This is a typical character of optical illusion."
"I conjecture that our brains interpret the image as the cylinder whose section is generated by a perpendicular cut, that is, the section is the intersection of the cylinder and a plane perpendicular to the axis of the cylinder. The actual edge of the cylinder is a space curve which is not embedded in a single plane."
We want to see the top of the cylinder as being a perfect flat cross-section because the brain is predisposed to right angles, Sugihara supposes in a separate paper on "anomalous mirror symmetry." This is illustrated by the Ames room illusion:
So, in terms of our cylinders, what we want to see is the image in the middle, even though the reality may be the image on the right.
Then, in using a mirror, it's possible to generate both the apparently conflicting images b and c at the same time. We see comforting squares while the world is in fact much more warped.