The image you’re projecting is focused in the air, but you can’t see it unless something reflects the light to your eyes. The moving wand reflects the light just as the screen does, except that the wand reflects the image piece by piece.
When this reflected light enters your eyes, it makes an image on your retina. Your eyes retain each piece of the image for about 1/30th of a second—long enough to let you put the pieces together to make a composite picture if the wand is moving fast enough.
Your eye’s tendency to hang onto an image for a fraction of a second is called persistence of vision. Persistence of vision occurs because the light detectors in your eyes, the rods and the cones, continue to fire electrical signals to your brain even after a very short pulse of light has come and gone.
The deformed images produced when you move the wand in the shape of a cylinder or cone are examples of a map projection. The planar image from the computer or slide is projected onto a curved surface. The resulting deformations are like those that occur when the spherical surface of the earth is mapped onto a flat map.
You might wonder if you could see the image by looking directly into the slide projector from the place where the screen used to be. The light in most projectors is too bright to try this experiment. However, the answer is no. Your eye can only make images of the light that actually enters the pupil. When you put your eyes where the screen was, light from only a small part of the image enters your eye, so you cannot see the entire image.