Mouth radical on the left, the indexing radical for any sound coming from a throat. It groups 啼 with 哭 weep, 叫 call, 唱 sing - the radical anchors the meaning in 'noise made by mouth', here a loud cry or animal call.
Right side 帝 supplies the sound, though dì has shifted to tí in modern reading. Don't read the 'emperor' meaning into 啼 - 帝 also appears in 谛 (listen carefully) and 蹄 (hoof), all sharing this phonetic series rather than imperial imagery.
