口 (the mouth radical) marks 呸 as an oral expression - specifically the sharp expulsion of breath used to express disgust or contempt, the spit-sound of bah! and pooh! Same family as 啊, 喂, 哎 and other emotive interjections.
丕 pī provides the sound, lightly shifted to pēi. The phonetic stem here is purely for sound - there is no semantic contribution from the literary meaning. Same phonetic feeds 邳 (Pī, place name) and a handful of other rare characters.
