I can't even understand the diagram: how is exhaling into a glass of water supposed to make breathing easier?

Breathing into water, per se, doesn't help. The air blown into the system is lightly pressurized, and this pressure helps inflate lungs just like inflating a balloon.

The water bottle is there because it's difficult to control the amount of pressure. Sinus tissues and lungs are very delicate, and pressurized air also messes with blood/gas ratios, so it's important to use the minimum pressure you can get away with. The bottle of water allows the pressure to be controlled simply and precisely, by adjusting the depth of the hose.

My understanding is that having water at the terminal side of the air flow creates back-pressure, which helps prevent the lungs collapsing.

Is this sort of system used for gunshot or stabbing victims where the lung was punctured?

Increasing the pressure in the lungs, by increasing the back pressure in the exhaled air.