To some extent every single sound in the world works the same way. There’s always something ‘vibrating’ (which really just means shaking!) and those vibrations shake the air back and forth.
Those sound waves travel through the air...into your ear and that’s how you can hear the sound.
The real question is ‘what’s vibrating?’. When you “throw” your Ultimate Paper Popper the cone shape catches the air and the paper flips back and snaps into its new position.
This causes the vibrations that make that awesome popping sound.