The system uses the "Henry Law" which states that the amount of gas that can be dissolved in a liquid is proportional to the pressure on the liquid. Raise the pressure - more gas can be dissolved in the liquid. Decrease the pressure - gas dissolved in the liquid releases the gas. This is exactly what happens when you open a can of soda; carbon dioxide gas is dissolved in the liquid and is under pressure in the can. Open the can, releasing the pressure, and the gas fizzes out.

Bodner's system apparently uses a centrifuge to lower pressure in part of a small amount of seawater taken into the system; dissolved gas is extracted. The patent abstract reads:

A self-contained open-circuit breathing apparatus for use within a body of water naturally containing dissolved air. The apparatus is adapted to provide breathable air. The apparatus comprises an inlet means for extracting a quantity of water from the body of water. It further comprises a separator for separating the dissolved air from the quantity of water, thereby obtaining the breathable air. The apparatus further comprises a first outlet means for expelling the separated water back into the body of water, and a second outlet means for removing the breathable air and supplying it for breathing. The air is supplied so as to enable it to be expelled back into the body of water after it has been breathed.

Human beings have been thinking about how to breathe underwater since they started swimming. This long-held desire plays an important part in one of the first great science fiction novels, Jules Verne's 20,000 Leagues Under the Sea.

It consists of a reservoir of thick iron plates, in which I store the air under a pressure of fifty atmospheres. This reservoir is fixed on the back by means of braces, like a soldier's knapsack.
(Read more about Jules Verne's diving apparatus)

(Check out this Functional Captain Nemo Diving Suit)

More recently, I distinctly remember an episode of the sixties sf series Voyage to the Bottom of the Sea in which a scientist decides that the best way to breathe underwater is to give himself gills. Alas, once equipped with gills, and fully acclimated to life in the sea, Dr. Jenkins and his associate lie in wait outside the submarine Seaview, converting every diver who emerges from the ship into mermen.

(From The Amphibians - aired Mar-08-1965)

And, of course, everyone remembers the scene in which intrepid Jedi Obi-Wan Kenobi and Qui-Gon Jin don pencil-sized breathing masks to explore the swamp lakes of Naboo in The Phantom Menace. This trick is used again in the most recent Star Wars movie.

(Hmm, perhaps those small cylinders are centrifuges...)