What to see
Start by watching the demonstration on the right. In particular, track the blue shape obscured by the orange squares.
After one cycle you know that there is a blue square of constant size which slowly rotates. But to nearly every observer it looks like the square is changing its size, “breathing” so to speak.
What to do
You can change the gap size manually. You can also change the transparency of the occluders (with the slider below the orange colour selector) and you may find a setting where one can “switch” between the breathing perception and a constant square size. Further controls are self-explanatory.
Usually, we are remarkably adept at detecting and identifying surface boundaries despite incomplete optical information. However, in “breathing illusions”, rigidly rotating surfaces appear to pulsate or deform, even though a straightforward process of geometrical interpolation across space would reconstruct the veridical surface boundary. It is generally believed that such nonrigid outcome depends on a failure to apply a rigidity constraint across spatiotemporal discontinuities (more: Bruno 2001).
The present phenomenon seems related to “motion binding”.
Shiffrar M & Pavel M (1991) Perception of rotation through apertures. Z Exp Psychol: Human Perception and Performance 17:749–761
Bruno N (2001) Breathing illusions and boundary formation in space-time. In: Shipley TF, Kellman PJ (eds) chapter 17, 531–556 (ISBN-13: 978-0444505064) [PDF]