How does it "work"?

Just because we can measure how "strong" the incoming sunlight is, how does the instrument enable us to determine the water vapor in the atmosphere?

The answer is a little bit involved and requires an understanding of the solar radiation spectrum (the "rainbow" colors shown by a light prism with a beam of sunlight passing through it) and how the atmosphere affects it.

Sun Photometer Graphs

Figure 1 below shows the Solar Spectrum, or the energy emitted by the Sun at different wavelengths in the visible and "near infrared region of the spectrum.  When this energy is received at the surface of the earth, the spectrum looks different due to the absorption by the atmospheric constituents and scattering back to space.   In the top half of Figure 1 the wavelength is expressed in microns (millionth of a meter), whereas in Figure 2 the wavelength is expressed in "nanometers" (one billionth of a meter), such that 2500 nm is equivalent to 2.5 microns.  Another unit of wavelength of visible light that is often used by scientists is Angstrom (10⁸ Angstrom = 1 meter, such that 10 Angstrom = 1 nm). 

Figure 1

 

Figure 2