You can look at vapor pressure like this. . .
Vapor pressure and the kinetic energy within a liquid go together. One doesn't exist without the other. The more kinetic energy within a particular liquid, the higher its vapor pressure due to the higher speed of the molecules. Can you see where I'm going with this? When this kinetic energy (directly related to vapor pressure) is greater than the pressure exerted by the atmosphere above it on the surface, it can push through.
If you don't already know, heat is kinetic energy. That's why heating liquids allows for the kinetic energy (which increases vapor pressure) to increase and overcome the atmospheric pressure; hence liquids boil with added heat.
Another way to look at it is picture a vacuum with zero atmospheric pressure. With this abscence of atmospheric pressure pressing down on a substance, it will readily boil away due to the fact the vapor pressure within the substance in the vacuum has no other pressure to compete with.
Hope this helps.