Fuel needs oxygen to burn... So all engines need oxygen, and get it from air. However, opening the engine to the environment Willy Nilly is a terrible idea fraught with danger to the engine internals, so the air supply gets filtered. I believe a CAI affords 2 improvements over the stock air filtration system: A higher flow of air, and ideally, colder air, which is more dense, so also equates to more air at once. More air means more oxygen which means you burn more fuel to keep the air/fuel mix in proper balance. Therefore, more air = more fuel = MORE POWER.
And we like more power. More power is good.