As you move the graph paper, light from the Mini Maglite spreads out equally in all directions. As the distance from the bulb to the graph paper increases, the same amount of light spreads over a larger and larger area, and the light reaching each square becomes correspondingly less intense. For example, adjust the distance from the bulb to the graph paper to 1 inch. At this distance, the graph paper touches the card. A single square (measuring 1/2 inch) will be illuminated. When the graph paper is moved 2 inches from the card, four squares will be illuminated on the graph paper. When the graph paper is moved 3 inches from the card, nine squares will be illuminated. At 4 inches, sixteen squares will be illuminated, and so on. The area illuminated will increase as the square of the distance. (If you are using metric graph paper, try moving the graph paper at 2-cm increments.)
The intensity of light is the power per area. Since the energy that comes through the hole you cut is spread out over a larger area, the intensity of the light decreases. Since the area increases as the square of the distance, the intensity of the light must decrease as the inverse square of the distance. Thus, intensity follows the inverse-square law.