The golfer gets on the downswing plane when he turns his hips to the left to initiate the downswing. The plane for the downswing is inclined at a shallower angle than the plane of the backswing, and its lateral axis points slightly to the right of the golfers target.
I don't see anything on page 88 about a change in spine angle resulting in the change in plane. The diagram shows no movement of Hogan's head and only the rotation and dropping of the arms to the point where the trail elbow gets to his hip to produce the shallower plane. This is the same movement of the hands I described in my post (with my addition of the description prior to the elbow getting to his hip).