You can think of a debt being settled and the slate wiped clean, so that a person can walk away with nothing left hanging over them. In court the debt is a charge, and wiping it clean means the person is free. Turn the same idea on yourself and it becomes the part you owed on the day: you gave it, and you can walk away from it too.
From the word’s history
The source traces acquit to Old French aquiter and Medieval Latin acquitāre 'to settle a debt', from ad- 'to' plus quitare 'to free' — a doublet of quit. Settling and freeing sit inside the word from the start.
來源資料說 acquit 來自古法文 aquiter 和中世紀拉丁文 acquitāre（把債務結清），由 ad-（朝向）加 quitare（使自由）組成，跟 quit 同源。「結清」和「放開」這兩件事，從一開始就在這個字裡面。
Wiktionary entry for acquit, etymology: Middle English aquī̆ten, Old French aquiter, Medieval Latin acquitāre (“to settle a debt”), from ad- + quitare (“to free”); doublet of acquiet, related to quit. (packet ref kaikki:3f6f69a190036ae4:527864149:51832)
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