FM transmitter design?
L1 and C4 are the parallel-resonant tank circuit for the RF oscillator, and Q1 is wired as a common-base amplifier that provides the gain required for oscillation, with the feedback (and output coupling) provided through C5.

In the common-base configuration, the input is applied to the emitter and the output is taken from the collector. The base is held at ground (at radio frequencies) by C2.

C3 bypasses the positive end of L1 to ground at radio frequencies, and it has negligible effect on the frequency.

R1 and R2 provide the base bias for the transistor, and R3 provides the DC path for the emitter current to ground.

R1, C1 and C2 form a bandpass filter for the audio signal. As you suspected, C1 blocks DC.

The audio signals modifies the base bias voltage, causing the operating point of the transistor to shift slightly. One effect of this is that the effective capacitance at the collector node varies slightly, causing the resonant frequency of the tank to shift with it, and this is what generates the FM signal at the output.