How to supply oxygen to a culture vessel?
Gases are supplied to culture vessels by design. Flasks, for example, may have filtered caps that allow gas in, or microplates with lids that have enough room under the lid to allow a gas exchange interface with the culture medium. Of course with these traditional culture vessels you need to make careful consideration about fill volume, since the gases won't penetrate 100% through the culture medium. 

In the case of your paper they cut a large hole in the side of the cup and plugged it with cotton, then fed a small plastic tube through into the cup. They attach a gas line and feed oxygen through the tube. The cotton plug is where the gas will escape back out without letting contaminants in, as the cotton plug acts as a filter. The system seems to have no other outlet, so it makes sense. 

In order to control a gas flow rate, you need a regulator of some sort to act as an output control.