How does the calibration panel Work ?

At a basic level, it’s called Calibration because it affects the initial conversion from the raw data to the RGB rendering. It was originally added so that if you thought the application’s conversion was a little off, you could “calibrate” it.

That function was more effectively addressed when Adobe added camera raw profiles. Today, if you think the initial rendering is off, you create a custom camera profile. But people still like to use Calibration for color correction, even though that isn’t what it was originally for.

I don’t have the color science background to talk about what Calibration does colorimetrically, but Jeff Schewe talks about it in Chapter 3 of his book The Digital Negative. He also doesn’t describe the exact color math, but he makes the important point that each color primary Hue and Saturation option actually controls the other two primaries: Red adjusts blue and green, Green adjusts red and blue, and Blue adjusts red and green.

(Jeff wrote The Digital Negative and The Digital Print inspired by Ansel Adams’ books The Negative and The Print.)

Some of the better explanations and examples I’ve seen about using the Calibration panel are by Blake Rudis on his f64 Academy channel on YouTube. That is probably because Blake has a much stronger practical understanding of color theory than others on YouTube.

Therer are plenty of YouTube demonstrations on how it works and how to make use of it. Therefore, I recommend that you use Google or YouTube's own search facilities. By way of example https://www.youtube.com/results?search_query=lightroom+calibration+panel