The 8-pin PCI-E connector has 3 12V and 3 ground wires. So, how can it deliver more power? The limit on the previous 6 pin connector was set to limit voltage sag, so that even at maximum voltage drop on a long PSU cable, the voltage at the card would remain satisfactory.
Under higher load, there is the risk that the voltage sag in the cables will be excessive and cause the voltage to drop out of spec under high load. The PSU regulates the voltage on its PCB, but under high load on long cables, this can be in spec, while the voltage at the end of cables is out of spec. The new connector provides a sense function, that allows the PSU to sense the voltage that is actually reaching the power circuits on the card, and a suitable designed PSU could regulate that voltage specifically. The 2 new pins are used for this sensing function.
The 8-pin connector adds 2 pins on the right of the 6 pin connector. The middle ground on the 6 pin (pin 5) is moved to 'top right' (pin 8). This allows the graphics card to sense if an 8 pin connector is used. If pin 8 is grounded, an 8 pin plug is used. If it's not-connected a 6-pin plug is used, and the card should operate in restricted power mode.
This leaves 'bottom left' (pin 4) and the now unused ground (pin 6) which can be used by the PSU to sense voltage. If the PSU doesn't have a remote voltage sensing function, these lines can be left unconnected. Pin 6, the original ground pin, can be left connected to ground if the sense function isn't being used.