I think the calculus-based explanation is the simplest one, and also easy to extend to an arbitrary number of dimensions.

If we look at the direction from the tip of the cone to the base, the volume is an integral of the area of the circles that are the cone sections. The radius of those circles grows linearly when the point is moving from the tip of the cone to its base. The area grows proportionally to a square of the distance from the tip. An integral of x^2 is x^3/3. Hence 1/3.