This is the first time I'm studying those subjects (I'm still in high school) and my teacher couldn't give me an answer. I'm referring specially to Newton's law of gravitation and Coulomb's law of electrostatics.
These are the long range forces; the simple explanation is geometric: if you model force as a bundle of arrows that pierce the surface of a sphere, or as dots on a balloon, then as the sphere increases in size, the density of the arrows (or dots) decreases as the square of the radius.
This follows because the surface area of a sphere is $S=4\pi r^2$.
The intensity of light or sound also falls off as the inverse square, and for the same reason.