Picture one process making one thing. Beside the thing everyone wanted, something else keeps coming out, without anyone planning it. That side thing is the by-product. Once you can see that picture in a factory, the same picture carries straight over to what people do: you act for one reason, and something extra falls out beside the reason.
From the word’s history
The word is literally by- plus product, and product goes back to Latin prōdūcere, to lead or bring forth. Read together, it says: the part that was brought forth beside the main thing.
字面上就是 by-（在旁邊、附帶）加上 product（產品）。product 這條線更早來自拉丁文 prōdūcere「引出、生出」。合起來可以讀成「在主要的東西旁邊，被一起引出來的那一份」，跟今天的兩個用法都對得上。
Wiktionary (kaikki extract), by-product: 'From by- + product.'; source_ref kaikki:3f6f69a190036ae4:1121978363:20964
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