put between, insert, jam, get caught, sandwich
挾 centers on holding something tightly between two surfaces: putting an object between, inserting it into a gap, jamming it in place, or getting caught in a pinch. The sandwich sense is a natural extension of being pressed between two things.
挾 combines 扌 (hand) with 夾, which likely contributes both sound and the idea of pressing from two sides. The overall structure suggests a hand squeezing or holding something between.
The left side is a hand 扌, and the right side 夾 looks like a person 大 with two small marks on each side, as if being squeezed. Picture a hand jamming something into a tight space between two objects.
For キョウ, imagine a hand squeezing a kyo (きょ) sound out of a squished toy: kyo -> キョウ, and the toy is jammed between the fingers.