As Kale explains, the project is based on a Colpitts Oscillator, which combines an LC circuit with a transistor amplifier for feedback. The frequency of oscillation is somewhere in the 100KHz range, which cannot be heard by humans. Enter the Arduino. When the trigger is pressed, an Arduino program translates the oscillation into an audible tone that is played out of the speaker. When the oscillation exceeds a certain threshold, it also emits a celebratory light show because… why not?
Kale walks through his entire build—along with the science of it all and how it works—in the video below. The schematics can be found on Imgur.