What does bile actually do to fat?

It emulsifies it - breaks large globules into tiny droplets so the enzymes that actually digest fat have vastly more surface to work on. Bile does not chemically digest anything itself; it is a detergent, and the chemistry is the same as washing-up liquid, with molecules that have a water-loving end and a fat-loving end.

The liver makes it continuously and the gallbladder stores and concentrates it, releasing a dose when fat arrives in the small intestine. That is why people who have had a gallbladder removed manage perfectly well but often struggle with very fatty meals: the bile still arrives, just not in a concentrated pulse when needed.

Its colour is a disposal system doing double duty. Bile carries bilirubin, the yellow-brown breakdown product of old red blood cells, which is how the body excretes them - and that pigment is what gives faeces their colour. When bile cannot get out, because of a stone or liver failure, the bilirubin backs up into the blood and the skin and eyes turn yellow, while stools turn pale. Jaundice and pale stools are the same fact seen from two ends.