The Escape Trunk is the hatch through which up to 22 sailors can exit a submarine, either for routine missions or in an emergency, at up to 600 feet of water. For emergencies, a deep submersible can attach to the deck directly above the hatch for deeper rescues.
The water pressure on the outer hatch is always greater than the air pressure inside the submarine, which prevents opening the hatch. Only when the pressure inside the escape chamber is equal to the sea pressure can the hatch be opened. Thus the compartment must be sealed off from the interior of the submarine and the pressure inside the chamber must be raised to sea pressure in order to make it possible to open the escape hatch.
The first video was shot aboard the USS Virginia and represents current technology.
The second video shows a Steinke Hood unrolled and points out pertinent details.
The third video shows a practice ascent filmed at the escape trainer at the Naval Submarine School, with cameras inside and outside the suit giving you the best idea of what it is like without actually doing it. A submariner from the Cold War period told me this was referred to as the HO HO HO drill, as it was required to yell that as you ascended to relieve pressure in your lungs. In this video they just yell for the same reason.
GROTON, Conn. (Dec 7, 2010) Naval Submarine School Senior Instructors Frank Gorham, left, and James McCloud discuss with contractors the scheduled maintenance of the school's 84,000-gallon capacity escape trunk in the submarine escape trainer. The winter service period provides a break in the pressurized ascent training both enlisted and officer submarine Sailors undergo as part of their initial accession or during their skills assessment and enhancement training. At right is the trunk in use.
Through the years there has been great progress in the development of equipment used to safely ascend from a submarine and survive once at the surface.
Escape Equipment:
The Momsen lung was a primitive underwater rebreather used before and during World War II by American submariners as emergency escape gear. The Momsen lung was invented by Charles B. Momsen (nicknamed "Swede"). Submariners would train in a 100-foot deep escape training tower using this apparatus. It was first introduced as standard equipment on P- (Porpoise) and Salmon class boats. The device recycled the breathing gas by using a counterlung containing soda lime to scrub carbon dioxide. The lung was initially filled with oxygen and connected to a mouthpiece via twin hoses containing one-way valves: one for breathing in and the other for breathing out.
The "Steinke Hood", invented by Lt. Harris Steinke was a an inflatable life jacket. It featured a hood attached to the top of the life jacket which captured air vented from the jacket under decreasing pressure so that the escapee could breathe during a long ascent. The Steinke Hood was designed and tested in 1961, when inventor Lt. Steinke escaped from the USS Balao. Further testing was done from a bottomed submarine at 318 feet off the Dry Tortugas. 
Current technology offers submariners a much higher degree of safety and survivability with the MK-10 Submarine Escape Immersion Equipment (SEIE). Several submarines have already installed the new system including Key West, one of 17 attack submarines home ported in Pearl Harbor. This suit allows survivors to escape a disabled submarine at depths down to 600 feet, at a rate of eight or more men per hour. The SEIE is designed to enable a free ascent from a stricken submarine and to provide protection for the submariner on reaching the surface until rescued. The assembly is comprised of a submarine escape and immersion suit, an inner thermal liner and a gas inflated single seat life raft all contained in an outer protective stowage compartment.. U.S. Navy photo by Chief Journalist David Rush (RELEASED)
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The suit not only keeps the escapee dry and protected from cold shock during escape, but also acts as a thermally efficient immersion suit on reaching the surface. Full protection is therefore provided while deploying and boarding the life raft. The suit provides sufficient lifting force to take the escapee from the submarine to the surface at a safe speed of approximately two to three meters per second.