Oil pulling is an ancient Ayurvedic medicine method. In practice, the oil pulling extracts bacteria from the mouth and body due to the chemical properties of the oil in a more natural environment. Mouthwash eliminates (or covers up, depending on what mouthwash you use) the current bacteria in your mouth. Oil pulling is basically sucking the bacteria from your mouth and body hence the term "pulling."
Oil pulling has an effect that is seen gradually as with most other natural healing methods.
Also, I am currently using the exact same mouthwash pictured below.