If you go to an Art or Design school. Seeing and producing illusions like this is one of the assignments that they usually will give you in a 2D design class.
As it has been described above, if you can concentrate (if school, we learn how to look at them by squinting as we would when discerning simple shape or color - or, if you have ever learned how to look at one of those weird 3D images made out of what looks to be paint splatter) on the two squares, then you will be able to see that they are indeed the same shade (not color, color is used to describe something else)
Ah, that worked for me. For people wondering how to do the technique to see "Magic Eye " images, you focus your eyes so that the image doubles and and overlaps the image. That causes a stereoscopic illusion when done on any things that overlap. You could practice it here. Focus your eyes so that the the first abc overlaps the second abc -- now you have three abc's in your vision, the 1st and 3rd abc are being seen out of one eye and the abc in the middle appears to be almost 3d.
a...b...c......................a...b...c
In this case, I could see that A and B are the same color by tilting my head and then focusing so I saw a double image of A overlapping B.
Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when my I look at this illusion and I interpret the data in such a way to determine distinct colors. My expectations of reality and the information being transmitted and received are not lining up. If they were, the illusion wouldn't be an illusion.
The number 2 is prime; the number 6 is not. What about the number 1? Prime is defined as a natural number with exactly two divisors. 1 is an illusionary prime if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1." Building on these bad assumptions could result in all sorts of weird results much like dividing by 0 can make it look like 2 = 1. What a tricky illusion!
An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the visual system is reporting RGB style colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, my visual system is distinguishing between something else entirely. Now that I have the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It merely highlights an obvious reality.
Addendum: This post was edited to fix a few problems and errors. If you are at all interested in more details behind the illusion presented here, there are a handful of excellent comments below.