Stare at the centre of the figure for a while. Some ‘scintillating’ activity will build up in the violet and blue annuli. Some observers also report a circular rotation within these regions; things will begin to “run around in circles”.
This image is called “The Enigma” (1981) by the Isralian painter Isia Leviant (1914–2006). It is (was? being presented in a permanent exhibition “Euréka”, Palais de la Découverte, Paris. [I have not been able to dig up more information on this since the people who had worked on this project are no longer with the Palais de la découverte.]
Mechanism currently unknown (as far as I know, but see Zanker 2004, PDF below), an ‘enigma’. Associated activity in the “motion area” MT (Zeki, 1993) is an interesting –if unsurprising– finding, but does not ‘explain’ the effect. I'm sure eye movements play a role.
Leviant I (1982) Illusory motion within still pictures: The L-effect. Leonardo 15:222–223
Zeki S, Watson JD, Frackowiak RS (1993) Going beyond the information given: the relation of illusory visual motion to brain activity. Proc R Soc Lond B Biol Sci 252:215–22
Leviant I (1996) Does ‘brain-power’ make Enigma spin? Proc R Soc London B 263:997–1001
Barch D, Kumar T & Glaser DA (2003) Modeling the illusory motion of Enigma with an excitable neuronal array [Abstract]. Journal of Vision, 3:72a
Zanker J (2004) Naturwissenschaften 91:149–156 [PDF]