A quagmire is a dangerous place, like the muddy shoreline of a pond. Because it's so hard to climb out of a quagmire, the word has also come to also mean any difficult or sticky situation you find yourself in.
Long ago, quag was a synonym for "bog" or "marsh," a swampy area where water seems to sit instead of drain out. Mire is another word to describe such a place. As a verb mire means "stuck," like someone who's mired in quicksand or mired in work — both prevent you from going anywhere. In a quagmire, you get stuck physically — or, even more commonly, in a situation that is hard to escape because there is no easy solution.