First, let's consider how those interesting longitudinal wrinkles arise:
Mahadevan's group showed, in "Wrinkling of an elastic sheet under tension," Nature (2002), and "Geometry and physics of wrinkling," PRL (2003), that the wrinkles arise from the inability of the film to undergo Poisson contraction at its ends, near the rigid supports (in the tape video, the hands). This constraint produces a stress state that shears the film into wrinkles with predicted wavelength $\lambda=(tL)^{1/2}\varepsilon^{-1/4}$, where $t$ is the thickness, $L$ is the length, and $\varepsilon$ is the applied axial strain.
Let's now consider the difference between peeling the film before and after implementing this wrinkled state. It appears that the level of tape-paper adhesion needs to be intermediate for the trick to be effective; if the adhesion is low, then peeling is effective, whereas if the adhesion is high, then the paper would be pulled off by the out-of-plane wrinkling deformation.
We appear to have a state where (1) some parts of the paper stochastically stick strongly to the tape and (2) the paper cohesion is similar to the paper-box adhesion, so that pulling at a small point can expand to a triangle of ripped and delaminated paper. In addition, (3) shearing the interface detaches the tape effectively.
Thus, if we pull the adhered tape from one direction (left image below), small attached areas expand into a big ugly rip, whereas if we selective delaminate strips through the buckling method (right image below), any small attached areas are localized, the detachment at the connected, sheared regions is clean, and the damage doesn't propagate:
I would expect this trick to fail if the adhesion is strong enough that shearing becomes ineffective in separating the interfaces cleanly.