纟 is the left-side simplified form of 糸 (silk). Embroidery is fundamentally needle and silk thread on cloth, so the silk radical is the perfect anchor. Compare 织 (weave), 编 (braid), 缝 (sew), 绒 (velvet) - all silk-radical fabric arts.
秀 supplies the sound, unchanged. The semantic echo is rich - embroidery is the elegant, decorative show on a fabric, exactly what 秀 implies. The pairing reads as silk-elegance. Same phonetic: 锈 (rust), 诱 (entice), 莠 (weeds).
