叼 specifically means gripping something in the mouth/beak without chewing - a dog carrying a stick, a bird holding a worm, a person with a cigarette dangling. The common pattern is 叼着 + object.
Mouth radical on the left does straightforward semantic work - 叼 is what you do with the lips and teeth, gripping something without using the hands. Sits in the same mouth-action cluster as 吃 eat, 咬 bite, 含 hold-in-mouth, 吸 suck, 吐 spit.
Right side 刁 supplies the sound directly - same diāo with no shift. The phonetic itself depicts something hooked or twisted, which faintly reinforces the way an animal hooks its prey in its teeth or a smoker dangles a cigarette from the corner of the lip. Same phonetic appears in 凋 wither and the surname 刁.
