Why do planes retract their landing gear?

An extended landing gear is one of the least aerodynamic parts of an aircraft: legs, wheels, shock absorbers, hydraulic lines and open doors exposed to an airflow of hundreds of kilometers per hour. That geometry produces parasitic drag, drag that doesn't contribute to lift and only slows the aircraft down. On a light aircraft with fixed gear, the penalty is accepted in exchange for simplicity, but on a commercial jet extended gear increases total drag so noticeably that pilots deliberately use it as an aerodynamic brake during rapid descents. Retracting it right after takeoff allows the aircraft to accelerate, climb with less power and burn less fuel throughout the whole cruise.

Drag increases with the square of speed, so the faster an aircraft flies, the more it costs to fly with the gear down. At an airliner's cruise speeds, around 800 to 900 km/h, extended gear would simply be unthinkable: besides the fuel burn, the aerodynamic loads on the doors and legs would exceed structural limits, which is why there's a maximum speed for flying with the gear extended, and an even lower one for operating it. Retracting it also cuts noise: the gear is one of the main sources of aerodynamic noise on approach, ahead even of idling engines on some aircraft.

At cruise, the outside temperature is around -55 °C and the air is extremely dry and thin. Tires, hydraulic seals and brakes fare better in the enclosed environment of the gear bays, where they also pick up some heat from nearby systems. After takeoff, the brakes are applied briefly to stop the wheels from spinning before they're stowed, preventing gyroscopic vibration from damaging the bay.

Retractable gear needs hydraulic actuators, position locks, sensors, doors and a gravity-drop emergency extension procedure for when the hydraulics fail. All of that adds weight and requires inspection. That's why many trainer aircraft, light utility planes and some regional turboprops keep fixed gear, sometimes faired to recover part of the lost efficiency. Retraction becomes worthwhile once cruise speed is high enough that the fuel savings comfortably outweigh the weight and cost of the mechanism.