Why Our Brains Are Fooled by Illusions

"The short answer is that our brains are programmed to see the world in three dimensions instead of two. "

Not necessarily. Different illusions operate on different perceptual principles. Some work on a lack of depth perception information, in which case that statement would be correct.

But others like the one above take advantage of our colour constancy - that we can percieve a colour to be the same under different levels and types of illumination.

Others work on how our visual cells are excited or inhibited by certain visual information such as the Hermann Grid illusion:
http://www.vashtie.com/blog/wp-content/uploads/2009/01/black-dots-optical-illusion.gif

...or the leaning tower illusion:
http://neurophilosophy.files.wordpress.com/2007/05/kingdom.jpg

More often that not, an optical illusion is our brain's way of trying to make sense of visually confusing stimuli.

Unlike what this article claims, psychologists specialising in perception (usually the ones who investigate these sorts of phenomena) do not ignore the larger picture, and do indeed talk about optical illusions not only on a neural level, but on an evolutionary level too.

And I'm not sure what kind of position "a brain research" is supposed to be.

I would like to make a comment on the last example in the linked gallery.
As I am red-green colourblind, I cannot with all my imagination see the "green" and "orange" squares; they look the exact same in the "before" and "after" pictures.

Any other colourblinds have the same experience?

Am I color blind? I can only see gray., Not blue on the Left, Not Yellow on the right.

They are only supposed to look blue or yellow in the above pictures. Blue/yellow colourblindness is extremely rare.

Are they still called "Rubik’s cubes" when they are 5 by 5?

Using pixie.exe, one can tell these "gray" squares are not completely gray. They are blue-gray, or yellow-gray. Very few of the pixels are actually gray; the majority of these surfaces are blue- or yellow-tinted.

Again, it's pixie.exe