The plastic or paper or tape or whatever, as far as I know, works because it keeps the heads in the reader a pretty uniform distance from the magnetic field of the stripe on the card, ironing out the little scratches and dents in the magnetic material.
Card readers have an automatic gain control (AGC) circuit that deals with the differences in stripe condition between different cards. Generally speaking, the older a card gets, the weaker its stripe's magnetism will be, probably just because of mechanical wear. The AGC auto-calibrates the reader to compensate for this. But if an old card's unevenly worn such that the magnetic material's strength varies substantially along its length, the AGC won't be able to keep up, and the card won't be readable.
If this unevenness in strength is caused just by the proximity of the head to the stripe on the uneven card, then wrapping the card with something thin and magnetically invisible and then swiping the card so that the card is pressed against the wrapping material should even out the changes and, thus, allow the card to be read. Which, it seems, is why the magic bag trick works.
