Answer: It is due to the low air pressure in the stopper and the atmospheric pressure.
Explanation: Whenever is there is a pressure, there must be a force exerted. Because,
Pressure = Force/ Area
So, when the bulb of the dropper is squeezed, the air pressure inside the dropper decreases because all the air is forced out.
When the dropper is placed in a liquid and the squeezed bulb is released, the air pressure outside the stopper surrounding the liquid becomes high and forces the liquid inside the dropper.
The rise of the liquid into the stopper is due to the atmospheric pressure.
This is similar to drinking liquids using a straw.