Did you look up "common-base amplifier"? The emitter is the input and the collector is the output. The stray capacitance from the collector to the emitter causes positive feedback. Any amplifier oscillates when it has positive feedback and some voltage gain that this transistor has.
I wasn't aware of this configuration. If I understand it correctly, the base of the transistor is common for RF signals due to C2 (capacitor at the base which is a short for high frequencies). The emitter is the INPUT while the collector is the OUTPUT. To keep the oscillations going from the tank circuit (formed by L1 and C4), it needs positive feedback. This feedback is provided by the stray capacitance (not shown on the schematic) between collector and emitter.
Some questions which do arise:
1) Does C5 (capacitor at the output) has other purposes than just providing AC-coupling to the (optional) antenna? If so, does it contribute to the collector-emitter stray capacitance? Is it correct that the it is not necessary to have the red wire drawn by JimB due to the stray capacitance?
1) Still figuring out how oscillation frequency is modified? I think there are two things to this:
(a) the common-base amplifier amplifies amplitude (I've read), it almost doesn't amplify current. So I think by coupling in the audio AC voltage, the magnitude of amplification is modified? Can I call this "the gain"?
(b) Still wonder how the FM is modified by the input voltage. I guess it has something to do with the some junction capacitance of the transistor varying by some voltage (hence act like some varcap)???
Well, this is a very interesting exploration for me into transistor amplifiers and FM modulation
Thank you all so far!