What to see
Follow the moving target with your gaze. Observe that the two oblique dotted lines undergo illusory motion.
What to do
Uncheck the checkbox “dotted”: The illusory motion of the oblique lines all but vanishes. There's lots more to play with: Changing the background luminance (lower slider) reveals that the illusion is strongest when the luminance is in between the black and white parts of the lines. Etc.
Ito, Anstis & Cavanagh published this striking effect in 2009. They explain in their paper (referenced below) “The reason is that the black/white contrast signals between adjacent dots along the length of the line are stronger than black/grey or white/grey contrast signals across the line, and the motion is computed as a vector sum of local contrast-weighted motion signals.” I couldn't have said it better myself :).
Somewhat simplified: When your gaze follows the target, the images shift on the retina. This motion is seen by your many many local motion detectors in your visual cortex. All of these only see a small hole (aperture) of the entire image, as schematised in the neighbouring figure for one motion detector. Given only the information from this small aperture, the motion signal by the black-white border is stronger than the horizontal shift, thus the local motion detectors decode an oblique motion which is at odds with the global picture, hence the illusion. Well, I'm not sure if that was any clearer…