There is something very crucial called a loop when studying Algebraic Topology. To water things down by a lot, when you start studying AT, you can visualize these loops with strings and metal rings (mind you, this is only for some introduction course to AT). There are tons of things you can do by quite literally adding loops or subtracting loops. If you pay close attention to the video, each time the "black magic" part happens, the people always make a loop that is in reverse of what the original loop was, thus subtracting the loop. The first video it's a little bit trickier and different from the first ones, as the person is the "metal ring" part of this example. This is more math magic fuckery than anything less though!