So it's important to call out here that they're using a modified LCD panel. A traditional LCD panel has two components: a backlight that constantly provides solid white light, and the LCD in front of it, which
darkens each pixel. This is why on your computer monitor, if you send it a "black" signal, you still see some light. The LCD is as dark as it will go, but some light still bleeds through because the backlight is on all the time.
In the paper, they're effectively using two backlight configurations. When displaying the solid object, they shine the backlight on the object, which scatters some of that light back towards the viewer. During this phase, they mask the image the LCD wants to display by having it render a greyscale image, where black pixels should hide the physical object, and white pixels should let the physical object shine through. In this mode, the physical object becomes the backlight, effectively.
In the image display mode, they point the backlights towards the LCD panel, and display their super-imposed image. In this mode, the object is not being illuminated, and so its contribution to the image is minimal. This is also why an enclosure is required, because the display needs to be able to throw the physical object into shadow during this second phase.
All of this is, of course, happening extremely quickly. The panel alternates between these two modes so fast that the human eye can't perceive the flicker, and persistence of vision simply combines the light contributions from both modes. The result is what you see in the images and video example.