Catholic seminarian here, not historian. The Red (scarlet, technically) of Cardinals represents the blood they pledge to shed for the faith. Fun fact, the birds were named after the churchmen, not the other way around. The sash, which is called a fascia, can be worn by priests as well (I even have one as a seminarian) and not only makes you look cool, but represents the purity they commit to. It is technically a type of cincture. The vesting prayer for a cincture is "Praecinge me, Domine, cingulo puritatis, et exstingue in lumbis meis humorem libidinis; ut maneat in me virtus continentiae et castitatis.", meaning "Gird me, O Lord, with the girdle of purity, and extinguish in me all evil desires, that the virtue of chastity may abide in me."

The zuchetto was originally to keep a tonsured head warm, but it is now just a symbol of their office (based on color).