Phantograms work by presenting the viewer with a pair of flat images precisely distorted to mimic the anticipated perspective of a three-dimensional object viewed from the phantogram's intended vantage point.
The anamorphic distortion of the source image crucial to the illusion can be understood by likening the images to projections of a 3D object onto a plane (e.g. a sheet of paper) originating from the location of the viewer's eyes.