When a substance's multiple phases are in thermodynamic equilibrium with each other the vapor pressure is the pressure exerted by a vapor existing above a liquid surface. Vapor pressure is related to volatility; the greater the pressure above the liquid the easier it is for vapor molecules to escape the equilibrium and transition to the gas phase.
A simple analogy would be to relate the vapor pressure to the vapor molecules' desire to break free from the equilibrium (multiple reasons for this, more complicated than the question though.) The force of the atmospheric pressure acts as a permeable barrier that can prevent the vapor molecules from escaping (by this I mean transitioning to the gas phase;) when the atmospheric pressure is greater than the vapor pressure the atmosphere is able to contain most of the vapor but when the pressures are equal the vapor is trying to escape just as much as the atmosphere is preventing it from escaping. This is the boiling point; when any more liquid in the equilibrium transfers to vapor the pressure of the vapor is too much to contain and molecules freely convert to gas.
Does that make sense to you? I can put together something more detailed and maybe file some diagrams/pictures that will help if necessary.
Edit: I think this image might help:
Pay attention to the arrows which indicate the opposing forces of the vapor pressure and the atmospheric pressure. When the temperature reaches the boiling point the force of the vapor pressure equals that of the atmosphere's and liquid can freely transition to the gaseous phase.
This is also good