To droop is to limply or loosely hang downward. Have you ever seen the long, floppy ears of a basset hound hanging down? Those ears are drooping.
When things droop, they're loose and hanging down. On a day without wind, a flag will droop. If you are sad or embarrassed, your head might droop a little. Evidently, drooping your head when you’re feeling down is something people have been doing for a long time. Droop is also related to the words drop and drip.