Lenticular images are the neato transforming pictures that often came on trading cards in the 1980s and 90s. They were handy for freaking out young children or filing your nails. Turn them one way and they show one picture. Turn them another and they show another. How? A trick of the light. And plastics.
Lenticular images are the kind of things they used to give out as free promotional material. They were best suited to things like trading cards of Transformers, because when looked at from one position, the card would display an image of the untransformed robot, while from another angle, it would display the image of whatever it transformed into. (On the back could be a description of why transformers transformed into cars with passenger compartments even when there weren't people to be passengers on their world.) The cards were covered with a piece of ridged plastic.
The images take advantage of light's tendency to bend, and only bend a certain amount. The ridges of plastic essentially 'block' parts of the image from the viewer. Light from certain parts of the image is reflected or bent away from the viewer. Each ridge, across the page, directs certain slices of the image back to the viewer. As the viewer moves, they are exposed to different parts of the ridges and see different slices of the page.
The image underneath the ridges is a series of interlaced slices - a little like a colored bar code. Each slice matches up with a section of ridge, and the slices come together to make the full image. Early lenticular images generally only had two pictures and flipped back and forth. More modern ones will be a little more complicated, with many different images, each corresponding to a different segment on the ridge. Some will even present a 3D picture, by showing slightly different image slices to each eye. For example the right eye could see one angle of a face, and the left eye could see another. This is how the eyes regularly build 3D images in the mind, and so the two images combine into a 3D picture. All it takes it the right kind of sectioning, and, of course, plastic.
Top Image: World Imaging
Poster Image: Wiki Commons
Ridge Diagram: Lenstar