If you turn a glass upside down and put it in a sink full of water you'll find the air stays in the glass and stops the water filling the glass. This is exactly how the undersea lab works. The lab is sealed apart from the opening in the bottom, so the air inside it can't escape, and the trapped air keeps the water out.
The air has to be at the same pressure as the water (otherwise the water pressure would compress it and allow some water in) so the pressure is higher than on the surface. That's why the chaps in the lab have to open a valve on the barrels containing the towels before they can open them.
You have to be very sure there are no leaks in the undersea lab. Any leak would allow the air in the lab to leak out and bubble to the surface, and sea water would rush in to replace the leaked air.