Well, the other answer frankly seems too complicated using statistical mechanics and all, let me give you a simplified version
Imagine a beaker, when you zoom in on the surface you see all the atoms and the molecules jiggle faster with more energy than the other atoms just below it. The vapour pressure is "almost" due to these atoms, by "almost" I mean that it is these atoms which get yanked off of the liquid and create a "commotion" at the top of the beaker.But there is a limit as to how many of the molecules get yanked of the surface which is dependent on the amount of heat you provide. Now take the beaker and empty its contents to another beaker which has more surface area.
If we zoom in on the surface of the new beaker we can see that there are more atoms at the surface which now "snap" out and create an even greater "commotion" at the top of the beaker, thus creating more vapour pressure.
Now according to the definition of pressure, increasing the area should decrease the vapour pressure but it doesn't, thus we conclude and say that it is not a surface phenomenon.