Heat is a measure of the average translational kinetic energy of molecules. Molecules exchange kinetic energy through collisions with one another. Just relying on collisions alone however takes awhile for the heat to spread -- one "layer" of molecules bumps another and exchanges a small fraction of energy. This layer bumps the next exchanging a small fraction and so on.
By blowing on it, you are effectively moving away the "bumped" layers quite rapidly compared to letting them move around on their own. So you are continually introducing a new source of cool molecules to absorb the energy.
This is far more effective than the conduction through the metal spoon (which acts on essentially the same principles and will depend on the convection within the soup on the spoon as to how effective it is). Unless your spoon is made of copper or something far more conductive than the stainless steel typically used for silverware.