As the amount of gain on a Cisco AP with dipole antennas remains the same irrespective of the power level of the radios, how come the coverage area increases with increase in power level?
By changing the transmitted power you change the IR. Any change in the IR or gain of the antenna will change the EIRP and thus the coverage area.
Antennae are passive amplifiers.
Bryan's explanation is right on but let me ask you another question.
How is it that I can change how far away I can hear my home stereo (the range) without changing the size of my speakers? I increase the power input to the speakers (turn up the volume).
If an access point transmits but there is no receiver to hear it, does it really exist? 8)
Increasing the power of the signal also increases its propagation characteristics. Les power, less propagation. More power, more propagation, and less loss before the signal gets to the receiver.