Airless pump bottles keep your skin care fresh by protecting them from too much exposure to air. When you depress the pump it creates a vacuum effect, drawing the product upwards. See the image below for a diagram of the whole process. 

Since a small amount of air in the bottle is unavoidable, you may need to shake your serum bottle and give it a few pumps to get the vacuum effect going the first time, and the process may need to be repeated periodically.