How does this FM transmitter work?
Q1 amplifies audio with a gain of approximately R4/R5. This audio feeds Q2 (an oscillator) and modulates its base-collector miller capacitance thus producing a varying output frequency. Q3 is an RF amplifier to provide more RF signal to the antenna.

That's the intention but there is a problem (as mentioned), Q1 won't work and indeed C4 should either be shorted out or possibly bypass R5.

C5 is to remove RF artefacts from Q2's base. At RF, the miller capacitor is in series with C5 and, if the power supply is correctly decoupled that series circuit (as far as RF is concerned) appears in parallel with CF6 (the tank capacitor)

C9 is positive feedback to allow Q2 to self oscillate.

The tuned circuit on Q3 is just filtering to ensure the FM output doesn't produce harmonics that might overlap onto other stations.