An easier way to understand it, without knowing the math, is to know how it's made. You play audio into a very similar device and it's needle scratches the grooves. When you then have a needle pick up the grooves it's moving the exact same way the needle was forced to move by the original.

It's similar to how a speaker and a microphone are basically the same device. If you take a speaker and plug it into a microphone input, it still works (though they're tuned differently so it's not as good). A microphone has a crystal vibrate, which creates an electric signal. If you play that electric signal into a crystal it vibrates and creates the same sound.

There's no math or anything being done for this to work. It's purely mechanical. It's just a copy of what the needle did when sound was played into it, so another needle running through it recreates the same sound. You can use math to represent it, but none is being done by the device (other than just the laws of physics).