Why do higher frequencies mean higher data rates, and why do we even need frequencies to begin with?
I don't know what do you mean by "presence/absence" of voltage, but in general, what you describe is exactly how it works:


High voltage on the data line is (usually) defined as logical 1
Low voltage is logical 0


The "frequency of the circuit" is just the maximal number of times the transition between voltage levels can happen in a second. 

Please note that if you send a long string of logical 1's, there is no need for these transitions at all - you can keep the data line constantly at high voltage. However, the question arises "how many 1's were sent during some period of time"? How do you count them if the data line is at constant voltage? For the purpose of resolving the above ambiguity, there is usually a "control" signal which determines the "pace" of the circuitry - the Clock. On each rising/falling edge of the clock signal, the voltage is sampled from the data bus and its value is checked for determining whether it is logical 1 or 0.

In the case of a circuit where Clock is present, the frequency of the circuit is just the frequency of the clock signal.

There are schemes when the circuit is not synchronized with Clock, but these schemes are much more complex and they are (usually) employed in special application which require very high bit rates.