Jack Russell in an invisible Skinner Box!
Operant conditioning is a type of learning that occurs when a behaviour's likelihood of being repeated is affected by its consequences. In this case, a treat is presented upon the closing of a drawer or the pulling of a shoe string, positively reinforcing its behaviour and motivating the dog to repeat it. Some shaping might have occurred before the dog could learn to do so in the first place. Shaping may involve a giving-withholding-giving treat pattern as the dog gets closer and closer to the desired behaviour. You can see evidence of shaping in the video, when the dog waited for a treat after taking her shoes off (the point at which during training/shaping, it would've gotten a treat in the beginning), and could only get actual treats after both shoes were taken off AND slippers were fetched! It's not animal slavery if they're happy doing it!
Just think though, all this work could easily be avoided if you'd just sing a song. Witness:
Snow White and the Seven Dwarfs (Cottrell & Hand, 1937)
Enchanted (Lima, 2007)