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I don't understand how this circuit works. What is the output of an electret microphone? Which area the transistors are in and how they drive leds? The microphone is an electret one called K193837BS(I didn't find his datasheet).

Assuming that we forget the missing resistor (in order to not blow the LED).

If the electret presents a small resistance, the tension divider electret-R1 is such that Q1 cannot conduct. All the current through R3 is at the base of Q2 and thus, Q2 conducts. The LEDs turn on.

If the electret presents a large resistance, an acceptable tension can be developed at the base of Q1 to turn it on. That deprives enough current from R3 to turn Q2 practically off. The LEDs turn off.

Capacitor C2 connects the electret only if the voltage at the electret changes.

That could thus produce LEDs turning on and off if there is a varying sound (music).

Not that Q2 is never "far" from being turn on or off, which makes it reacts fast (to ... relatively ... slow change).

Run the "DC SWEEP" to see the effect of the resistance of the electret on the current through the LED.

As the resistance of the electret increases, Q1 turns on harder and harder, starving Q2 from its current. Q1, in deep saturation, can have its Vce as low as 0.2 volt (even less for some BJT). That same voltage is the Vbe voltage for Q2, which should be around 0.6 volt to turn Q2 on.