Remember that pressure is just force per unit area. You could even draw a free body diagram for a molecule floating above the surface in its phase equilibrium state.
In an equilibrium gas molecule above a substance, there is the force of atmospheric pressure pushing down on the molecule and the vapor pressure pushing up. If vapor pressure is less than Patm, the molecule remains in this equilibrium state. But, when we reach Patm, the substance boils. At lower Patm, there is less force to overcome and the substance boils more easily. Conversely, the higher the vapor pressure, the lower the boiling point.