Left mouth radical - pictograph of an open mouth. 叽 is an onomatopoeia for chirping, squeaking, or muttering - sounds coming from a mouth or beak. Same family: 哇 wow, 嘿 hey, 啊 ah, 嗡 buzz - every imitative noise word picks up this radical.
Right 几 supplies the sound - jǐ drifting to jī with tone change. 几 pictures a small table or stool. Pure phonetic; the table meaning has no link to chirping. Simplified 叽 picked this lean phonetic to replace the much heavier 幾 of traditional 嘰.
