There are two basic reasons why increased voltage lets the caps get charged faster and the transistors switch quicker. First of all, the increased supply voltage increases the drive current across the transistor, so it charges it's own output cap quicker.
Secondly, all voltages are increased, so the input voltage is increased as well. In my crappy diagram I idealized the input into a perfect square pulse, but in reality, the input is coming from another transistor's output itself, so the input also rises and falls for some given period of time, like below, so increasing the input voltage at the gate allows the transistor to start switching faster as well.
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