When someone swallows a drink and claims it "goes down the wrong hole" what really happens?

Aspiration. The trachea (carries air in and out of the lungs) is situated in front of the esophagus (carries food and drink to stomach). Usually when you swallow, the tongue, palate, and vocal cords work to block entry to the trachea and nose and push food past those structures to the esophagus. Sometimes these mechanisms don't work completely and people aspirate ( food/liquid goes into the trachea).
The Trachea. When you swallow, usually the epiglottis (a flap of cartilage) will cover the trachea (a tube that you breathe through) so food and liquids do not obstruct your breathing. Sometimes the food or liquid can "miss" and accidentally pass down the trachea instead of the esophagus (the tube that leads to your stomach) causing choking or an unpleasant sensation.