This is classic animal training, with the snap of the finger as the positive marker for the food, and the voice as the negative marker for the tap of the stick. Keeping the pigs a little hungry is part of the training -- it keeps motivation up, same as it does for a dog or a hawk.
Below is much the same scene played out with 120 hounds and a whip. The marker here is not sound, but the placement of the whip. Down and swinging is "no," and up and still is "go".
Why not use a leash on 120 hounds and five pigs? Reasons! Not only would it be unworkable, but the timing would be horrendous.