What Is the Algorithm Behind Photoshop's "Black and White" Adjustment Layer?
Each (color) image is composed of RGB components. when you add (or reduce) a constant value to all pixels only in RED components you will see the effect equivalent to moving the RED tab towards the right, and same way reducing the RED component by a constant will have the reverse effect. 

Like wise you can increment/decrement each component by a fixed value as described. If you increase/decrease ALL RGB components by a same value this will be equivalent to change in brightness (basically you are adding/removing WHITE color). 

The Cyan, Blue, Magenta - corresponds to such transformation in CMYK color space. (But i guess, Blue in this color space corresponds to mix of Cyan and Yellow. So that's a bit tricky. The transformation is essentially same for all. 

The last element Tint: { Hue and Saturation } corresponds to same operations but here, the images is first transformed in HSV model and then HUE and Saturation is added/subtracted independently.

I don't know about the exact relationships of markers of the dial to the corresponding numbers but that you can figure out by trying out practical values.