The fuel in your engine requires oxygen for combustion. However, the air that enters your engine needs to be clean and free of contaminants, which is where the air filter comes in.
This filter (located in the air filter box) traps the dirt and debris that flows in through the bottom of the car. While clean air will continue uninterrupted into the engine, contaminants will be captured underneath the filter.
Without the engine filter, these contaminants will negatively affect the performance of your engine.
One unscrupulous way that some mechanics make money is by pushing for an air filter replacement when the current air filter is in good condition. This may deter car owners from replacing their air filter altogether.