C1 shorts 100MHz. The reactance is about 1.6 ohms. At audio frequencies, say 1 kHz, its reactance is 100000 times higher and the effect accordingly smaller.
ADD due the comment:
At audio frequencies (below 20 kHz) C1 causes a little attenuation, but the effect is so small that the audio signal well makes the operating point of the transistor to swing. Thus the internal capacitance of the transistor changes and causes FM.
Between 20kHz and 100MHz the effect of C1 increases gradually as frequency increases. But there's nothing happening between the audio band and the transmitting frequency in this circuit. At the transmitting frequency C1 has so low reactance (about 1,6 ohms at 100MHz) that you can think the base being grounded and that's needed to make the oscillator to work.
It's very common in electronics that the same circuit work totally differently in different frequencies at the same time. Mostly this is caused by frequency dependent reactances. The effect can be wanted as just in this case. Unfortunately many parts also have limited usable frequency range due unwanted reactances.
ADD2: DC source is a short circuit in the equivalent AC circuit. You must have separate equivalent AC circuits for 100MHz and audio due their big difference. In Audio AC circuit the 100MHz tank circuit coil is a short circuit and those few pF capacitors are open (=left out). In 100MHz AC circuit few pF capacitors and L1 are essential, the 1nF capacitor is a short circuit.
BTW. Your common base amplifie principle diagram belongs to the trashcan. Its taken from 1940's. BJT's of today work a little differently than early transistors. Unfortunately the tradition to use the same teaching image still continues. You can see the same in many explanations of radio waves. The sign of shit there is a pendulum, resonant circuit and gradually opening gap of the capacitor. Shit = nice story, unfortunately not physics.