I think I have an explanation, though I thought of it myself so I can't prove it.
Imagine you have a solid, movable barrier (i.e. a piston) that interacts with your liquid and vapor, which are stored in a sealed cylindrical container. On the other side of the piston is your atmospheric pressure. No other gases are in the cylinder. It's easy to see that if the equilibrium vapor pressure is equal to the outside pressure, the liquid-vapor balance will never change. If the eq. vapor pressure is GREATER, then the piston will just keep moving out until all the liquid has vaporized and the gas has expanded to equate the pressures. This is because keeping the piston in equilibrium requires the pressures to be the same. Yet if the pressures are the same, the liquid will be able to vaporize a little, making the piston move. The piston motion expands the gas, decreasing the pressure but allowing the liquid to vaporize even more. Conversely, if the eq. vapor pressure is SMALLER, then the piston will move in until all the vapor has condensed. Thus, the liquid boils only when the vapor pressure >= external pressure, for the case of a solid cylinder.
But what about the case of a liquid open to the air? Surely the liquid can vaporize easily from the surface, since the vapor always diffuses away? Yes, it can, and the liquid is constantly evaporating. But to liquid that is beneath the surface...the liquid ABOVE acts as a solid piston! Think about it: if the solid piston mentioned earlier were composed of the same material as the liquid, the scenario would be the same; The only criterion for a good piston is that it is impermeable. So now if the eq. vapor pressure is low, this "liquid piston" will press all the way down, as usual. If the eq. vapor pressure is high, then the "liquid piston" will expand all the way until everything boils. (We neglect the effects of surface evaporation, of course.)
See where this is going? The liquid state is essentially many, many layers of liquid pistons. When the boiling point is reached, these pistons all go from a state of being compressed to a state of continuous expansion. That is, when the vapor pressure equals external pressure, layers of vapor can form. Of course, there are details, like the weight of the liquid above contributing to the pressure on the liquid below, but hopefully this shows that theoretically, the relationship you were wondering about holds true.
Any questions? Counterarguments?
