According to China Radio International, chopsticks probably evolved from the use of twigs to pick up hot food. Relatedly, Fuchsia Dunlop's blog post on chopsticks recounts a memory of a camping trip in Sichuan where her guide cut and peeled some twigs from the trees to make chopsticks for their dinner.
Gong Dan's Food & Drink in China describes how in
the Zhou dynasty (11th-3rd century BC) chopsticks were used for eating meat and vegetables, while rice was still picked up with the hands. (Note, however, that pulchritude points out in comments that this may not be quite accurate.) These days, of course, rice is also eaten with chopsticks (assuming you're eating from a bowl — if you're given rice on a plate, often the most sensible way to eat it is with a fork and spoon).
Gong Dan also describes the etymology of the word. During the Zhou dynasty, chopsticks were known as 箸 (zhù). However, this is precisely homonymous with 住 (zhù), which means "to stop, to cease", and 住 was a taboo word aboard ships, since stopping a ship en route was considered bad luck. This problem was solved by referring to chopsticks as 快子 (kuài zi), a combination of 快 (kuài), meaning "quick", with the particle 子 (zi) as a suffix to make it into a "proper" word. Later, the bamboo radical (⺮) was placed above 快 to make 筷, since chopsticks are commonly made from bamboo, giving the modern word 筷子.
As someone who (a) didn't grow up using chopsticks and (b) was mildly teased at school for holding my knife and fork the "wrong" way, I'm reluctant to lay down any pronouncements about the right way to use chopsticks, but my preferred way of holding them is to lodge the bottom one firmly in the web between my thumb and index finger, resting it on my curled-in ring finger, and then to pivot the top one independently, pushing up with my third finger and down with my second finger as required, steadying it with my thumb the whole time.
I did find a pretty good YouTube video demonstrating this, but I've unfortunately managed to lose the link. There are lots of "how to use chopsticks" videos on YouTube, but be warned that some of them show rather suboptimal methods. Once you're holding your chopsticks in a way that you find comfortable, check out thorfinn's Chinese chopstick tips for what to do next.
Note also that chopsticks differ between cultures. Japanese chopsticks have pointed ends, while Korean chopsticks are made of metal and are flat rather than rounded in cross-section. Chinese chopsticks have blunt tips, and may be made from bamboo, wood, plastic, or less-common materials such as porcelain. I personally like the bamboo/wood ones because I find them more "grippy" than plastic ones.
The use of chopsticks to eat with is intimately connected with the way food is cut prior to cooking. Since there are generally no knives on the dining table, the cook must be careful to cut pieces of food in such a way that they can be picked up with chopsticks. This doesn't necessarily mean that everything must be bitesize — see for example this eGullet thread on ingredient sizing in Chinese cooking – but it's certainly something that must be borne in mind.
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