A cold air intake does not improve gas mileage, but it does increase a car's overall engine performance. Besides adding a cold air intake, turbochargers and superchargers also increase airflow to the engine.
Engines powered by gasoline work by taking in air and mixing it with gas. Burning this mixture results in the power needed to run a car. Adding a cold air intake cools the air sent to the engine. This helps because cooler air is more dense, allowing even more oxygen to fit within a smaller volume.
The more oxygen the engine gets, the more power it produces. Racers often pour ice over their intake manifolds before a competition to give them a temporary increase in horsepower. However, as the manifolds heat back up during the race, the extra horsepower is lost.
Despite having an increase in horsepower, cold air intakes do not decrease gas mileage by much. This is why cold air intakes are often falsely associated with providing better gas mileage. When a car gets a boost in horsepower, it usually consumes more fuel. Cold air intakes are cheaper and easier to install than superchargers and turbochargers as well. However, it should be noted that both turbochargers and superchargers provide a car with more power than a cold air intake.