There's really no way to put this delicately. Disgorge is just a fancy word for "throw up." Usually, this somewhat icky word is applied to birds or animals rather than people.
Gorge is an old-fashioned word for the throat, so to disgorge is to eject something through the throat. (Yuck.) In addition to its literal meaning, disgorge also has the broader metaphorical meaning of "to empty out." So a bus might disgorge its passengers and a spy might disgorge her secrets — for the right price.