The concentration is actually lower than you described because with Cyanuric Acid (CYA) in the water, the active chlorine concentration which is called hypochlorous acid is really only at around 0.05 ppm or even somewhat less, though the 3 ppm FC is a total capacity or reserve of chlorine. The active form of chlorine is fairly reactive and it looks a lot like water as you can compare it
here to water
here. So it's able to get into cells of bacteria, viruses and algae fairly easily and able to react with the chemicals inside such cells to kill them by disrupting normal cell processes (see
this thread for more info).
As for keeping the water clean, it's able to essentially "chemically burn" (known as oxidize in chemistry) some chemicals including the main ones in your sweat and urine, turning them into carbon dioxide and nitrogen gases, and water. When it's all done, the chlorine becomes chloride which is part of regular sodium chloride salt. For every bather-hour in a pool, it takes around 2 fluid ounces of 6% bleach to get rid of the bather waste so most of the chlorine you add to the pool gets used up by other means -- mostly getting broken down by sunlight and by chemically reacting with other things that may get into the pool.
Even though numbers such as 0.05 parts per million may sound very small, there are over 2 million trillion (2x10
15) of these active chlorine (hypochlorous acid) molecules in one teaspoon of pool water. The number of collisions of molecules in solutions is very high even at such low concentrations. Water and chlorine molecules in an 85ºF pool are traveling at speeds of around 1450 and 850 miles per hour, respectively. Even with the active chlorine so dilute, there are roughly 6700 collisions per second for each such molecule running into another on average (the frequency of running into water is much, much higher and I'm using the term "collisions" very loosely since solutions are much more complicated than gasses).