Think of it based on volume. Say you have a unit of volume of size X and you double it. It is now X times X times X which is 8 times more volume that the force has to effect. So, visualize a volume of one liter of water and add a single drop of food coloring. Then get a volume of water of twice those dimensions in each direction so you have a volume of 8 liters of water. Now add a single drop of food coloring to that and you will see that the color is much lighter than in the 1 liter example. Inverse square force is analogous to that.
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