That's a nice video. If you dont' have that good of a connection to have the horse go around you at liberty, the same can be done on a lungline.
The point being to have the horse turn , put an ear and an eye on you, walk such that he tracks with the inside hind reaching well under the belly, and that he does not drop the inside shoulder (the last horse in the video, the chestnut, was dropping his inside shoulder and crowding her a bit, IMO).
The horse cannot really bend his rib cage, per se. What happens is that when the inside leg steps under and acroll the belly , it pushed the ribcage to the outside , making the horse's body a bit "shorter" on the inside. So, in effect, there is a bend in the whole boy, supported by the inside hind taking a lot more weight onto it. It's that taking on of weight that characterizes "engagement".