Takeoff and landing distance have a lot of variables.
One is certainly thrust. The plane is designed such that it has enough thrust to reach takeoff speed within the available takeoff distance. For landing, the propellers are able to reverse the thrust and add extra braking power to stop in a shorter distance.
Another variable is landing and takeoff speed. If you lower takeoff and landing speeds, it takes less distance to get going and to stop. Being a propeller plane with a straight (non-swept) wing, it is designed to fly slower than jet aircraft. This means the plane needs less angle of attack to fly at slower speeds. The C-130 also has very large flaps for this purpose, and they contribute to lowering takeoff and landing airspeeds in two ways. First, the flaps are very large, which adds more wing area and also deflects more air downwards, increasing lift. Second, the flaps are placed directly behind the engines, which allows the flaps to direct some of the engine thrust downwards, further increasing lift.
Weight is also important. While the C-130 is designed for good performance at 155,000 lb, and can fly at up to 175,000 lb, it will be much lighter in a demonstration. From an empty weight of 75,562 lb, fuel and crew will still leave it well below its typical operating weights. The lower weight means that takeoff performance will be increased by better acceleration and a lower liftoff speed. Landing performance will also be better with a lower landing speed and better deceleration.
The newer C-130J also performs better than the older C-130H. The newer engines on the J model produce up to 4,637 shaft horsepower (shp) each (though they are listed here at 4,705 shp), while the engines on the H model produce up to 4,590 shp each. The empty weight on the J model is 75,562 lb and on the H model is 75,800 lb. So with the empty weight going down by 238 lb and the power increasing by at least 47 shp, and improved propellers, the J model has better performance. The takeoff distance at 155,000 lb is 3,127 feet for the J model versus 3,586 feet for the H model. The H model can take off in 1,400 feet at 80,000 lb (probably closer to the weight in demonstrations), so the J model can probably do even better.