I interpret the inverse square law as being the direct result of the area of a circle scaling as the square of the radius. Imagine dividing a given circle overlaid on a city into fixed sized area units. If you randomly visit a unit, the probability of visiting a particular unit scales as 1/area.

Note that this derivation is essentially identical to showing that electric field falls off as 1/(r^2). In that case the area refers to the area of a sphere through which field lines must pass.