The 555's oscillation is controlled by the capacitor from pins 2 and 6 to ground, and the resistor that comes from pin 3. When it is oscillating, the voltage on pins 2 and 6 swings between two voltage thresholds, that are set at 2/3 and 1/3 of the supply voltage by an internal voltage divider inside the 555. Pin 3 is high (roughly +V) and the capacitor charges up to the 2/3 threshold. When pins 2 and 6 reach the 2/3 threshold, the 555 flips its output low, and the capacitor discharges to the 1/3 threshold, when the 555 flips its output high, and the cycle repeats. The time taken for the capacitor to charge and discharge creates the frequency that you hear in the piezo.
When voltage is present at the circuit input, the small diode holds pins 2 and 6 close to the supply rail, and even though the 555 drives its output low and tries to pull pins 2 and 6 towards ground with the resistor, it can't, and so it can't oscillate.
When the input voltage disappears, pins 2 and 6 are free to be pulled upwards and downwards by the resistor, and the circuit oscillates.