'starch' (and thus 'starched') ultimately originates from Old English, specifically the word 'stearc' meaning 'stiff'.

'stearc' from Old English passed into Middle English as 'starch' meaning the substance or the quality of stiffness; the verb 'to starch' (to stiffen with starch) developed later and produced the past form 'starched'.

Initially it referred to 'stiffness' or the substance that causes stiffness; over time it came to mean 'to make stiff with starch' and, adjectivally, 'stiff or formal in appearance', which is its modern usage.