The video just shows how it could be done with cables (supposedly 1mm cables). Hmm. Well a 1mm cord would have a cross sectional area of 0.25*Pi mm^2 = 7.854e^-4 m^2. Stress = force/area. Lets assume David Copperfield weighs 165 lbs and his female partner weighs 125 lbs. That is a combined weight of 585.134157 kgs , and a force of 12642 N after gravity. Thus, the stress would be (12642 N)/(7.854e^-4 m^2) = 16MPa...so the wire would have to be made out of a material having a tensile strength greater than this. This is pretty low as far as tensile strengths go (Btw if there is a mistake in that calculation let me know, I just googled up the equations, and its not intuitive to me why the tensile strength is independent of cord length.)
What I don't like about this explanation is that it does not explain why people do not see the cords. Especially with lights being around, I would think that would be extremely easy to spot. If I can spot a fiber of spiderweb from 30 yards easily, and that is 0.15 mm in diameter, why can't people spot a 1mm cord?
They must be coated in some highly non-reflective material...perhaps some of those new "invisibility" meta materials