An intuitive answer has to do with the fact that a unit volume is defined by a "box" of unit side length. While a sphere with unit diameter would touch the wall

An intuitive answer has to do with the fact that a unit volume is defined by a "box" of unit side length. While a sphere with unit diameter would touch the walls of the box, the box has an exponentially large number of corners (2^dimensions) where the sphere has no presence. That is why the volume of a unit diameter sphere is much smaller than that of a unit box in high dimensions.