What to see
In the animation above you see “pigeons” craning their neck while scurrying along, and worms shrinking and extending while worming their way. Looks a little silly…
What to do
As you probably already guessed: the movent is, in reality, quite smooth. Try deselecting the “Grid” checkbox: now all scurrying gives way to fluid movement.
It is interesting to play with the colour of the figures (initially a dark grey). With bright colours the scuttling is less apparent, and there is a (rather light) grey level where the scuttling and motion in general is minimal. The illusion becomes stronger if you don't look directly at it.
This was Jun Ono, Akiyasu Tomoeda, and Kokichi Sugihara's contribution to the 2014 “best illusion of the year” contest. As these authors write, it is a version of the “kickback illusion”, which is closely related to the Stepping Feet illusion – see my explanation there.
A simple concrete explanation: Start with a “worm”, imagine it had the same colour than the dark grid stripes. Thus, when its front end crosses the dark stripe, this edge is not visible. The length of a worm is chosen so that when its front is on a dark stripe, its back will be on a light stripe. Consequently, when the front end is not visible, its back end will be obvious, and its movement will be apparent: the worm seems to shrink. As soon as the front end leaves the dark stripe, it extends again.
For the pigeons the same explanation holds. Their heads are one stripe ahead of the body, thus the nicking motion ensues.
Ono J, Tomoeda A, Sugihara K (2014) “Pigeon-Neck illusion” entry to the 2014 illusion contest
Howe PDL, Thompson PG Anstis SM, Sagreiya H, Livingstone MS (2006) Explaining the footsteps, belly dancer, Wenceslas, and kickback illusions. JOV 6,12:5