What is the area under the tongue called and why would it be swollen?

Floor of mouth: The most prominent structure in the floor of the mouth is the openings to the ducts for the submandibular glands. Sometimes the saliva glands can get infected or obstructed with a stone and swell. This is most common with dehydration. If painful or there is a lump there see an ent. If slightly swollen could try increased fluid intake and sucking on hard candy.

Glands: The floor of the mouth is loaded with glands that produce saliva and mucous, as well as numerous big blood vessels. One could be damaged or blocked. See a dentist or oral surgeon for an evaluation.