Why A phase-locked loop compares the phase of a reference signal with the output of a voltage-controlled oscillator, filters that error into a DC control voltage, and steers the VCO until the two stay locked. Insert a programmable divider in the feedback path and the VCO must run at N times the reference, which is exactly how a synthesizer generates many frequencies from one crystal. Feed an FM signal into the phase detector instead, and the control voltage the loop generates to keep up with the deviation is the recovered audio, so the same circuit demodulates FM.
Watch out The choice about comparing two digital signals and counting pulses describes digital logic like comparators and counters; a PLL does compare phase, but it is an analog feedback loop, not a pulse counter, and power amplification is not a PLL function at all.
PLL = phase detector + loop filter + VCO: divide the feedback for synthesis, tap the control voltage for FM audio.