One place where branches do appear is within the graceful lilly-like arms of the crown. Food sticks to mucus in the ambulacral groove on the inner side of the branching tentacles. Then the animal's nervous system can control the arms to bring food to the mouth, located in the cup of the armored Calyx, where it travels though a digestive tract until it is excreted from the nearby anis. Instead of muscles, the arms use a "hydrostatic skeleton" of fluid-filled tubes to control their motion.
However -- the branching fossil of the picture does not conform to the usual way in which crinoid arms branch from a common joint. Sometimes crinoids can feature side branches called Cirri, but these generally branch off the side of a single columnal.
The answer was suggested by Crinoid expert Joe Koniecki. That pattern of branching is more typically seen with the anchored holdfast, and our fossil above likely shows where the crinoid was rooted into the ocean floor. Thanks Joe for helping us solve our Puzzle fossil!
For more great photos of crinoids, be sure to visit Joe's crinoid website.