Air intakes are basically in every car, their main purpose is to suck in Air or specifically Oxygen, which in turn triggers combustion in the engine. No oxygen means no combustion.
Now what after market Air intakes do is they provide better air flow, they don't have Resonators like stock cars do, and pod filters are large in surface area and allow more air to enter the engine, no resonators means no restriction and no restriction means high air flow, this leads to better throttle response.
Cold Air Intakes usually run longer in length and suck in Cold air, which is rich in Oxygen compared to hot 'dirty' air inside the engine compartment, this leads to better combustion and thus better Acceleration and throttle response.
They don't effect the fuel consumption much, but yes when you hear the engine grunt, you'll tend to floor the pedal, it's chemistry, that would obviously affect fuel consumption, as long as driving style is acceptable, fuel consumption won't be affected much.
Hope this helped