Air conditioners use blower fans to transfer the warm air away from a desired location and return cold air. Contrary to popular belief, air conditioners don't really produce cold air; they merely transfer warm air away. The purpose of a blower fan is to maximize air flow not only through the machine, but also airflow of cooler air out of the machine.
Within the Machine
Heat will always move from low to high because heat rises. With an air conditioner, the heat rises and as it does the air passes through coolant coils transferring the heat out. The more air which flows through the coolant coils, the greater the efficiency of the air conditioner. In this way, blower fans work to aid the heat transfer by sucking the warm air out of the room to blow over the coolant, maximizing the quantity of airflow.
Outside the Machine
As the air cools within the machine, it condenses and does not transfer well. By using a blower fan, the air is not only moved across the coolant tubes faster, but warm air is expelled outside, and cooler air is returned to the room at a larger rate. If the dense air was left to transfer itself via pressure, it would be highly inefficient.