Think of a bar of soap in wet hands. You rub, and rub, and a thick white foam rises up and covers whatever you are washing. That foam is a lather, and the key part of the picture is that it does not pour out ready-made — it is worked up by rubbing. English keeps that same picture outside the bathroom. A horse that has run hard is rubbed by its own harness until its sweat turns to white foam, and that is called lather too. From there the picture moves to people: someone who has got themselves worked up over something is said to be in a lather.
From the word’s history
Wiktionary traces the noun to Old English lēaþor, a substance used for washing, and on back to a Proto-Indo-European root meaning to wash or bathe. The verb comes from the matching Old English verb līeþran, to soap or wash. So the washing picture sits in the word from the very start.
Wiktionary 記載，名詞 lather 來自古英文 lēaþor，指一種拿來洗東西的鹼，再往前可追到印歐語根 *lewh₃-（洗、沐浴）；動詞則來自對應的古英文動詞 līeþran（塗肥皂、洗）。也就是說，「洗」這個畫面從一開始就在這個字裡面，今天的泡沫義是它最直接的延續。
Wiktionary (kaikki extract kaikki:3f6f69a190036ae4:610730709 noun ety1; :610755808 verb ety2), entry 'lather'
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