The chip not only generates the AC feed, it senses changes in the load
due to the RFID chip modulating its tuned circuit.
The tuned circuit of the reader itself "transforms" the voltage up automatically
to the maximum the available power and the Q of the circuit allows - assuming
component values have been correctly chosen.
The way to increase range is choose a larger coil (range is proportional
to the size of the coil) and increase power to match. Also having the most
sensitive receiver helps
With a larger coil you get less signal (proportionally) from the RFID as
its cutting a smaller proportion of the flux from the coil, so more sensitive
receiver is needed to match, I believe.