So Plait identifies the illusion as being that these dots aren't really rolling around like it looks like, they're just moving in straight lines back and forth in a way that creates that illusion. In a way of thinking about it, he's right, but, if he thinks that means we can't say that the dots aren't really rolling around, he's wrong. Why? Math.
I don't need to get into any details of the mathematics here because someone else, John Baez, already did it (long before this video was made) and made a very handy website that not only explains it but also illustrates it. See that here, like really, do.
As Baez illustrates, what we're witnessing is a particular presentation of a special case of one circle rolling around inside of another, that case where the smaller inside circle is exactly half the diameter of the larger one that it's rolling around inside. By the quirks of trigonometry, if we take a circle half as big as another circle and roll it around the rim of the bigger one, the points on the diameter of the smaller circle move along diameters (straight line segments) of the larger circle.
This video presents an invisible circle with eight equally spaced points highlighted around its circumference rolling around the inner circumference of a circle twice as large. This isn't totally abstract either--the idea dates back almost a thousand years to the invention of a device known as a Tusi couple that takes advantage of this trigonometric fact to convert rotational motion into oscillating linear motion, where the visible dots could represent pegs on the teeth of a gear. (There may be a lesson here about being able to construct viral "illusion" videos using engineering-practical mathematical results that most people just aren't aware of, but I digress.)
In other words, this isn't an illusion, really. It's two phenomena at once, depending upon how we want to look at it. On the one hand, we have a group of dots that we can organize as being on the rim of a circle half as small as the one they're rolling around inside--so we really do have the eight dots rolling around the inside of the larger circle like it appears. On the other hand, we have a group of dots moving back and forth along eight diameters of the circle in straight lines (which happen to be equally spaced because the dots are equally spaced).
Neither interpretation is right or wrong, but the one that takes the object inside to be an (undrawn) circle rolling around the inside of a larger one is probably in many ways more mentally economical because it requires us to keep track of very little whereas keeping track of eight moving dots whose speeds change in specific ways (according to trigonometry) while being initially arranged in a certain configuration seems to require a great deal more effort.
Again, though, I really urge you to check out Baez's explanation because it shows more general cases that are actually a lot cooler than this one. Also, don't take it out on Plait. The most important bit is that he recognized that it is actually something cool going on (and was able to talk about mathematics related to it that ties into his speciality, astronomy, in a cool way, including the ever-awesome Spirograph toy that I also greatly enjoyed as a kid).