Take a look at this FM transmitter diagram:

The process is relatively straightforward:
1) The microphone takes the voice in.
2) This voice signal goes into an audio processor in order to create an input signal.
3) The input signal is combined with a carrier frequency that is generated by the voltage controlled oscillator (V.C.O.).
4) The result is a signal that carries our information but is not sufficiently powerful enough to be transmitted up through an antenna. The signal is amplified through two different stages:
5) The exciter stage amplifies the signal power up one level of output.
6) The power amplifier drives it up to another power output level, now sufficient for the transmitter’s antenna.

The receive process is very similar:

1) The transmission is captured by the receiver antenna and fed into an RF frequency selector.
2) The transmission goes into a mixer where it’s combined with a carrier generated by a local V.C.O.
3) The result is then fed into an IF (Intermediate Frequency) signal amplifier. It reduces the frequency through a filtering process that’s required to strip out the carrier more easily.
4) The signal is demodulated.
5) The result is fed into an audio section which amplifies the results so that it can be fed into a speaker and heard by the user.

An FM transceiver simply combines an FM transmitter and an FM receiver in a single unit. The receiver section of the transceiver is the same as what is seen in the FM receiver section and similarly for the FM transmitter.