When the bulb of a dropper is pressed, air present in the tube and bulb escapes out in the form of bubbles. This reduces air pressure inside the dropper.
As there is atmospheric pressure acting on the surface of the liquid, the pressure just outside the nozzle tip is higher now. The atmospheric pressure pushes the liquid from a high pressure region to a low pressure region - inside the tube.
This is how a dropper works.
Thus, atmospheric pressure is responsible for the rise of water in the dropper.