draw, reel silk from cocoons, elegant compositions
繅 centers on drawing out silk threads from cocoons, a process of reeling. The sense of elegant compositions likely extends from the idea of drawing out refined, thread-like lines of thought or art.
繅 combines 糸 (thread) with 巢, which likely contributes the sound and relates to the idea of a nest or cocoon-like structure from which silk is drawn.
The left side 糸 is thread, and the right side 巢 looks like a nest or cocoon. Picture drawing fine silk threads out of a silkworm's cocoon, reeling them into an elegant strand.
For ソウ, imagine the soft sound of silk being drawn: 'sow' as in sewing, but with a long vowel — sow -> ソウ. The thread is reeled out with a gentle, continuous motion.