This is a type of illusory motion (or motion illusion) called a peripheral drift illusion:
Peripheral drift illusions, as their name suggests, create the illusion of motion in the periphery of the visual field. Similar images that generate illusory motion in the center of the visual field have been termed central drift illusions. Both of these types of illusions use high contrast "luminance gratings" - repeating patterns of high and low brightness - to generate illusory motion.
The cause of motion illusion in general is not well understood, but research suggests that there may be slightly different reasons for the different types of motion illusions.
The dominant explanation for peripheral drift illusions in particular is that high-contrast colours have slightly different processing latencies at the retina, and as a result, neural signals arrive at the visual cortex at slightly different times, causing the illusion of motion (eg, Faubert & Herbert, 1999; Backus & Oruç, 2005). Comparison of brain activity using fMRI between actually moving and apparently moving images shows similar activity in the visual cortex, supporting the assertion that the illusion is caused at the retina rather than at the visual cortex (Larsen et al, 2006).
Since the difference in processing latencies is effective only at image onset, maintenance of the illusion should be enhanced by small eye movements (eg, saccades, fixational eye movements, microsaccades, and blinks). This is indeed what experiments suggest (eg, Otero-Millan, Macknik, & Martinez-Conde, 2012; Kuriki et al, 2008).