How do the positive feedback increase the gain of a regenerative receiver?
In a regenerative receiver the signal is amplified, then fed back into the input where it is amplified again (and again, and again, and...). Just before the point of oscillation the total gain is almost infinite. If the input has a tuned circuit then frequencies further way from resonance will be amplified correspondingly less, so selectivity is improved. 

The problem with a regenerative receiver is that when you add feedback it becomes more sensitive to any change in circuit parameters, increasing the risk of breaking into oscillation. At very high gain, just moving your hand closer to certain parts could be enough to set it off.

A related design that got around that problem was the Reflex Receiver. It amplified the RF signal just once, then sent the demodulated audio signal back through the same amplifier. Because the signal being fed back was at a lower and unrelated frequency, it didn't reinforce the original RF signal and so wasn't prone to oscillation. Gain was only doubled, but this was worth doing when the amplifying device (valve or transistor) was much more expensive than all the other components needed to complete the circuit.