How Does a Spectrograph Work?

This is not a particularly well written article - and what you see above is almost all of the article. The linked article includes one additional sketch, which is only so-so anyway.

It is possible to have a diffraction grating that transmits light, but I don't know of any cases where this type is used. All the ones I've ever seen are reflection type. With the reflection type diffraction grating, there is no change in the speed of the light. The grating makes use of the principle of constructive and destructive interference to separate light into its component wavelengths.

Most of the diffraction gratings I've seen (and I've done spectroscopy for years) are built into the curved mirror. Older spectrographs (or ones with an extremely sensitive detector) move the spectrum past a slit so the detector sees only a narrow slice of the spectrum at any given instant. When detecting extremely low light levels (due to a faint source and/or a widely spread spectrum) a very sensitive detector is needed, so the mechanical scanning method is used.

Newer spectrographs often use a CCD array to detect all the wavelengths at once, where each spot of the CCD array is seeing only a small portion of the spectrum. This type is much faster than the mechanical scanning type.