Monsters have traditionally, and still do, dissolve to dust when they die, making the harvesting of their meat difficult. The Hunger made the monsters clever, however... and they have long thought up a way to feed themselves without the meat turning to dust in their mouths.
They simply eat their prey alive.
Harvesting the meat for later works the same way. They strip their unfortunate hind of its flesh as it lives and breathes, and can then store the meat away while the creature expires (or, in the case of the Nice Cream Guy, is fed his own innards to regrow his flesh for harvesting again).