If there were not molecules above the liquid and space for them to occupy (a vacuum), then you would expect molecules to go from the liquid to the vapor phase. If the temperature of the liquid were equal to the boiling point, then the pressure of the vessel would reach atmospheric pressure (because that is what the boiling point is).
So molecules will go from the liquid phase to the vapor phase depending on the temperature of the liquid and the pressure of the vessel. However, our vessel is subjected to atmospheric pressure, so the total pressure will be the partial pressure of the ethyl acetate in this case, and the gases making up the atmosphere. If the equilibrium is shifted so no ethyl acetate molecules are present in the vapor phase, they will not return to the liquid phase. If nitrogen is passed over the liquid, we can alter the equilibrium and the partial pressure of ethyl acetate will be virtually zero and this will result in the ethyl acetate evaporating.
If you cap the bottle, an equilibrium will be created between the atmosphere and the ethyl acetate. The gas above the bottle is not all ethyl acetate though it has a strong odor, the amount in the gas phase depends on the vapor pressure and the temperature. If you leave the cap off, the ethyl acetate will diffuse from the bottle and eventually the liquid will evaporate. Blowing nitrogen into the vapor accelerates the process.
I'm not sure this is the most succinct explanation. I was trying to connect something you know or would expect, to the problem at hand.