Why does increased voltage increase processor performance
A CPU is, at its root, an extremely large sequence of FETs driving each other in complex combinations. These FETs have a very large number of possible states, only some of which are valid. Each clock cycle, the processor should move from one valid state to the correct succeeding valid state. The FETs are arranged such that they begin to switch from one state to the next only on a clock edge. But if one FET drives another, that means the driving FET has to complete its transition before the load FET can even start!

This is propagation delay, and the longest chain of propagation delays in the system determines your maximum clock frequency. You have to be sure that by the time the next clock edge arrives, all FETs everywhere have completed whatever transitions are going to occur. Otherwise you end up with your chip in an invalid or incorrect state, and all bets for correct operation are off.

So the minimum clock period (and thus maximum frequency) is a function of the longest propagation path in the chip, and of how long the individual transistors in that path take to switch. You can't change the propagation path once the die is etched, but you can change the time it takes for a transistor to switch. As with any FET, the switching time is affected directly by the time it takes the gate capacitance to charge. Since the capacitance is fixed, a higher voltage through the same FET results in more current flow, and a faster rate of charge. So increasing the rail voltage of the processor can increase the switching speed.

Of course, the down-side is that at higher voltages, the switching loss of the FETs also goes up, more than the gains from faster switching times make it go down. So higher voltage results in increased operating temperature, which can also affect the switching time of the FETs, and ultimately result in damage to the chip.

Also, quantum wibbly-wobbly effects can make the chip behave unreliably.