Why do baboons need red bums?

It is to attract mates. Females mate more frequently with males with brighter, more colorful rumps. There's a study in which the researchers painted the genitals of baboons in Tanzania and studied the female responses; they found that the more turquoise the more the females were attracted. Furthermore, in geladas (closely related to baboons; sometimes called gelada baboons), the amount and intensity of the red marking on the chest is directly proportional to testosterone levels, so it is certainly in the realm of possibility that the rump coloring is also related to testosterone levels. So from an evolutionary perspective, if coloring corresponds to testosterone, a male with more coloring is likely more capable of defending a female and her offspring, making them more appealing mates.