The mechanics of aquifers cause them to drip (pour in previous versions) down onto the level immediately below it. That level is the first non aquifer supporting one, which usually is stone (and some kinds of stone support aquifers), but it can also be clay.
All of these tiles are marked as wet, so normally you'll get a warning, but there are (at least) two exceptions: when digging upwards (typically by digging a staircase, the warning only appears when the damage is already done, i.e. you've dug out a tile beneath the aquifer, and if the tile is visible, it's marked as wet, but you get no warning, as DF assumes you've seen the warning.
When it comes to Stormfeather's problem, my first approach would be to follow Bumber's advice, i.e. to remove the aquifer tiles above and install walls along the edges. It's also possible to fill up the problem room with walls or just wall it off to let it fill up with water.
You can clean up the visuals by building a floor or constructed road on top of the muddied floor (and, when you're at it, construct nice patterns by using different color building materials). Note that DF does not support mud removal, neither as a "clean" task or an order, and the build floor/road on top and then remove process to remove mud is subject to a material transmutation bug, whereby some of the tiles can become soil ones. Magma can sometimes burn away mud, but that's a lot of work, and it can start fires in cave moss growing on top of it, resulting in ash on the floor, and the usual issues with dorfs and fire.