Automatic
The automatic color calibration finds known stars in the image and makes them look like their known color. To do this, Siril must know which part of the sky is seen in the image, so a plate-solving (also called astrometric solution) is required. Then, for each of the main stars of the image, color information (the red, green and blue relative magnitudes) must be acquired. For these two functions, an Internet connection is required, as an online database is queried.
Then, factors are computed for each stars and the average is applied to make red and blue channels match the level of the green channel.
Since star magnitudes are used for plate-solving, the image must be in linear mode.
The photometric color correction window includes the plate-solving step, which can also be done alone. The two tools look very similar. See the documentation on plate-solving.