Funny you should ask! I spent a few years on picking apart this particular illusion. It was the topic of my dissertation!https://digital.lib.washington.edu/researchworks/handle/1773/25444
Here's a broad-strokes summary of my explanation.
* There are two sources of visual motion, the circles themselves, and the gratings contained within each circle. If you look closely, you see that the spots move, but the black and white stripes also move within each spot.
* In the red ring, the stripes move in the same direction as the spots (relative to the overall picture, not relative to the spots.) In the yellow ring, stripes and spots move in opposite directions.
* The stripes and the circles are examples of "first order" and "second order" motion, respectively. In first order motion, a light feature in one frame (like a white stripe) moves to and matches a light feature in the next frame, or a dark spot changes position but still remains a dark spot. Most visual animals (at least as complicated as a fly) detect first order motion motion very well.
* However, if the visual system just matched light stripes to light stripes and dark stripes to dark stripes, you wouldn't see the movement of the spots. To see the spots move, a second mechanism has to detect that regions of high contrast (containing both dark and light parts) is changing its position. And humans have separate mechanisms that detect this more complicated type of motion.
* Your brain devotes a lot more matter to processing imagery right where you are looking, than to images coming out of the corner of your eye. When you view an object out of the corner of your eye, the information that gets through visual cortex has been "compressed," to borrow a computer term. So as you switch from looking directly at the yellow spot to viewing it peripherally, you are switching from an, uncompressed to a compressed view.
* If the compression was analogous to just blurring or pixelation, you'd expect to still see the circles moving, because they are larger than the gratings, and the gratings would blur out first! But the compression is more complicated than blurring.
* As it turns out, the compression the brain uses in peripheral vision accounts for first order motion but not for second order motion! Seen from the corner of your eye, the motion of the black and white gratings makes it through, but the fact that this motion was associated with the spots, and the different movement of the spots, is lost.
I'm not sure who made this gif, but it looks like it was adapted from a version of this illusion that Arthur Shapiro made a few years ago. Dr. Shapiro has also studied this type of illusion. Here's his blog entry from then:http://www.illusionsciences.com/2008/12/rotating-reversals.html