How is FM modulation is achieved using a single BJT
L1 and C1 form a resonant tank circuit - this produces the main frequency of the oscillator. Positive feedback from collector to emitter via C2 ensures oscillation quite close to the natural resonant frequency formed by L1 and C1.

The other factor is the reverse voltage between base and collector. There are components missing on your diagram that bias the base to a nominal voltage of about 2V and that reverse voltage to the collector defines the value of the transistor's internal miller capacitor. This miller capacitor is voltage dependent and with the base voltage variations this capacitor retunes L1 and C1 in sync with the audio signal applied to the base. Hence, the base voltage frequency modulates the oscillator.

It's not a great FM modulator but it does work providing the AC signal at the collector isn't too big in terms of volts peak to peak.