How does the component "alarm boost three pin inductor" make a piezo buzzer louder?
It works like an autotransformer. It's a transformer without galvanic isolation, primary winding is also a part of secondary. The purpose of using a transformer is to step up the voltage, thus increasing power.




simulate this circuit &ndash; Schematic created using CircuitLab
$$V_{piezo}= V_{signal}\frac{L_1+L_2}{L2}$$
Beware that there is a maximum on pulse duration, before the inductor saturates. Beyond this time the inductor will loose the inductance and it will become like a short circuit. Therefore you would have to find out which is the minimal frequency (in case you use a square pulse of 50% DT) that you can use this circuit.