I’m sure one of the resident physicists will be along to explain in more detail shortly, but it’s basically due to the fat changing the refractive index of the paper. Every time light moves from one medium to another it bends a little. For example when light moves from air into water it bends slightly, which is why objects underwater look like they are in different place to their true position. However the same is also true of when light moves form water into air or from air into glass or any other medium. What that means is that the more times light has to move form one substance to another on the way form an object to your eye the more times it gets bent. That’s why snow is white and yet water and ice is clear. Snow consists of lots of tiny ice particles with air tapped in between them. Light passing through snow has to change from air to ice and back again millions of times. As a result it gets hopelessly bent and scattered. The eye can still seethe light that has passed through snow just as well as light that has passed through water or ices, but the light isn’t reaching the eyes in straight line so it doesn’t form any coherent image. As a result we just see snow as translucent, while water and ice are transparent. Paper is approximately the same as snow. It’s made up of millions of little particles with air or glue spaces in between them. Light passing through paper has to pass from air to paper and back again many times, and on the process gets bent and scattered. So paper appears white and translucent for the approximately the same reasons that snow appears white and translucent. When paper is greased the oil soaks into not only the paper fibers but into the air spaces between the fibers. When light tries to pass through greased paper it only has to pass through one medium: greased paper. It never has to move from paper to air. As a result greased paper becomes transparent just as ice is transparent. Note that this is a very simplified explanation. There are other factors at play, notably the relatively close refractive indices of greased paper and grease itself, which is why grease works better than water. There is also the loss of specific wavelengths in the transition form one medium to the other that results in more light passing through greased paper as well as the light remaining coherent. But we can get into that once we’ve sorted out the basic effect of refraction in a complex medium.