有 and 机 are some of the first characters we learn in Chinese, but only recently did I find out that they mean "organic" when put together. Can anyone explain why this is?
As you may know, "Organic" in your case is originally a chemical term and its noun-"Organ" means "a collection of tissues joined in structural unit to serve a common function" from Wikipedia. Once the chemists incorrectly believed that "organic compound" could only be produced by lifes. I think this is how "organic" comes from "organ". I don't find the exact origin of the word "有机" in Chinese, but I believe it originates as "organic" originates in English, from chemistry.
机 originally means
主发谓之机。—《说文》The thing that controls shoot is called 机. 《说文》is an old dictionary.
trigger on a bow. A structure used to shoot the arrow.
Some other meanings extend from the original one.
critical part(point) in a system
organ. organ is a critical part in the body
mechanical structure. The critical part that controls the movement of a machine.
In recent times, after Chinese touched the sciences from the west, some pioneering scientists tranlated the terms in sciences. In general, they translated them in a liberal way.I think this is why “有机" means "organic", as you have "organic" from "organ" in English.
I have thought about this before, but from the perspective of Japanese.
Q: Why does 有機 mean organic? A: Because the fertilizer/feed that is used to raise whatever it is is made from stuff that works biologically rather than just chemically. Soooo, yeah... poop.
So, let us imagine a timeline of word formation.... *wo0Ooo0o~
機（mechanism）⇒ 機能(mechanism-ability = function) ⇒
生活機能(life-function) ⇒ 有生活機能(having-life-function = organic) ⇒
有機(having-life/organic) ⇒ 有機肥料(organic fertilizer) ⇒
有機栽培(cultivation with organic fertilizer) ⇒ 有機食品(organic foods)
So, 有機食品 is just shorthand for 有生活機能肥料栽培食品.
If that's true... then 機 by itself would mean a machine like a wooden weaving loom and by extension, "mechanism" as in "an assembly of moving parts performing a complete functional motion, often being part of a large machine".
Are you having any of that then?
If so, let me go back a little further in pure speculation...
糸(thread) ⇒ 玆(threads),
人(person) + 戈(spear/pole) ⇒ 戌 ＋ 玆 ⇒ 幾(loom),
幾 ＋ 木 ⇒ 機(machines usually made of wood - mechanism)
Yeah? No? Vapid Layman-Guesswork you say! (well I beat you to it, so sticks and stones...)
ALSO! I found that Waseda University has at least two copies of the oldest surviving book with the word 有機 meaning organic in it. I perused it to see where it's mentioned first in the book and of course it was on like next-to-the-last page. This book（舎密局開講之説）is a translation of lectures given by a Dutch scientist named Koenraad Wolter Gratama made by his assistant-professor Shousuke Misaki（三崎嘯輔）at a place called the 舎密局(Bureau of Chemistry) in Osaka in 1869.
The reason this is hilarious is because the characters 舎密 were chosen by the Japanese founders to approximate the sound of the Dutch word for chemistry, "chemie".
I never gave thought about how "有机" came to mean "organic" before.
I search in a dictionary give the following meaning for "机": 有生命的生物体器官的作用 (The functionality of organs in lifeforms). 《现代汉语词典》 the following meaning: 生活机能 (ability to live). "有机" is formed from this meaning of "机". But the usage of this meaning of "机" is very limited, probably only in "有机" "无机" "机能", so it should be save to remember these words as a whole.
BTW: I don't know how "机" got this meaning.
Actually, I have asked myself the same question some time ago. I find the explanations that take the word "Mechanism"（机制）into account as a reference to biological systems when trying to explain how 有机 is connected to the meaning of "organic" rather good. However, when I asked my Chinese teacher about it she told me that it also might have to do with 机 as a sound as it sounds similar to the way we pronounce "C" which is the chemical element of carbon which determines wether a compound is organic or not.