how about just looking at the red edge of the mobius strip on the first page of your video. you see a closed curve that crosses itself once. i.e.anything drawn on a flat screen is already projected into 2 dimensions, it requires some visual imagination to see it as in 3 dimensions.
You can actually do the gluing physically . Take a hollow square and do the needed gluing of edges , with the flip needed on the gluing for the vertices, i.e., if you do a "straightforward" gluing gives you a cylinder and one where you flip will give you the Mobius strip. To be more pedantic, both Cylinder, Mobius strip are quotient spaces of the square, i.e., spaces obtained by identifying sides of the square the right way.