When would somebody hold the stick like shown, and why?

Beginner helicopter pilots often tend to grip the controls too tightly, which leads to over-controlling and stick stirring.

Holding a pen or pencil so the middle two fingers are held away from the stick makes it unpleasant to hold the stick tightly, and the student learns to fly with small amounts of pressure.

Source: Shawn Coyle, The Art and Science of Flying Helicopters, page 37, figure 5.2.