I understand that 零 originally meant rain, however for some reason it eventually became fragments then also zero. How did the meaning of this character shifted to become also fragments (rain drops, maybe?) and, then, to mean also zero?
零 hadn't meant 0 in the ancient Chinese. It just because 10X reads 一百零几 -- means more than 100, so 零 filled up the position of 0 when the 0 conception came into Chinese.
google ＂零＂词源说明 get e.g. https://baike.baidu.com/item/%E9%9B%B6/36714 零，是形声字，古同“泠”，清凉。从雨令声，令为领省，即暴雨之前零星的雨滴，《说文》，“零，余雨也”。疾雨曰骤，徐雨曰零，久雨曰苦。引申出零落、零丁等义。古文中零非数词，近代以后方假借为数词。