People panicking breathe too fast, and breathe out more co2 than your body is used to. This makes you feel weird (light-headedness, fingers tingling, that sort of thing).

The idea is that re breathing the air you just breathed out should slow the co2 loss and make you feel better. My understanding of why we stopped doing it is its just a lot more effective to get the person to slow their breathing by talking them through it, and the paper bag thing never practically worked that well (while the theory is sound, in practice you have to strike a reasonably fine balance in having a rebreathing circuit open enough to allow plenty of oxygen in but still retain co2, and that's just really tricky).