The illusion works by manipulating the spatial frequencies of the image. The low spatial frequencies (the rough, fuzzy shape information of an image) from one image (e.g. Mr. Angy) are combined with the high spatial frequencies (the sharp edge information of an image) from a different image (e.g. Mrs. Calm). When viewing this hybrid image at a close distance, our perceptual system is able to extract and process the high spatial frequencies and thus we see the image where the high spatial frequencies were taken from. When viewing the hybrid image at a distance further away, our perceptual system can no longer extract the high spatial frequencies meaning we only see the low spatial frequency information, thus we see the other image. You can also get a switch in the hybrid image by squinting.