I stand to do this, but it depends on the size/weight and length of the bird.
The left hand would hold the bird's legs, around the middle of the shank or around the hocks, wherever feels more secure. Usually the fore finger goes between the legs so that the thumb and second finger supply the grip required to anchor the bird against the pull of the other hand.
The right hand, as has been mentioned, has to secure the head and, with a pull, as if to stretch the bird, and a degree of rotation, causes the neck to be dislocated behind the skull. The skill is in rotating the head on its dorsal axis (in effect bringing the top of the skull toward the neck), whilst persuading the vertibrae to part against the pressure from the encircling digits.
There is a tendency for the outside of the despatcher's leg, just above the knee, to come into play as an extra lever against which to create the stretching of the neck.