anneal, quench, temper
淬 centers on rapidly cooling heated metal in liquid to harden it: annealing, quenching, or tempering. The meanings all describe the same metalworking process of controlled cooling.
淬 combines 氵 (water) with 卒, which likely contributes the sound and the idea of suddenness or completion, reflecting the rapid cooling process in metalworking.
Water 氵 on the left and 卒 on the right: imagine a soldier (卒) plunging a hot blade into water to quench and temper it.
For サイ, picture a sigh of steam as the hot metal hits the water: sigh -> サイ, and the blade is annealed.
polishing one's sword; improving one's talents or knowledge