So let's go back to Bronze Age Shang China, three thousand years back, the time of the earliest chopsticks. What chopsticks survive are of bronze or bone, and they already closely resemble modern Chinese chopsticks. In itself, however, this tells us nothing about whether people actually ate with chopsticks. This is because they could just as well have been used for cooking instead of eating. If anything, cooking was more likely than not the original purpose of Chinese chopsticks, if simply because chopstick-looking tongs and stirring sticks are far more common globally than true chopsticks used to eat.
What, then, did the people of Shang eat with? The spoon and especially the hand. But because early Chinese cooking involved a great deal of boiling and stewing and brewing, chopsticks (really just sticks, at this early point) were important for stirring and tasting the food, or to use as pincers to take out or put in new food and ingredients without burning your hands. So there was already an important place for chopsticks in the Chinese kitchen even before people were using them to eat with.
So how did chopsticks become a tool for eating, not cooking?
The earliest Chinese texts generally take as granted that you'd be eating with your hands or perhaps with spoons. The Commentary of Zuo has an anecdote about a certain Prince Song of Zheng, whose index finger moved in anticipation whenever he saw something delicious and who tasted stew with his finger, in the year 605 BC. The Book of Rites also suggests:
Do not roll steamed grain [飯 fan] into balls...
Which implies that you'd be eating cooked grain with your hands, since otherwise you wouldn't be able to roll them into balls in the first place.
Same goes for meat:
Meat that is wet [and soft] may be divided with the teeth...
Later Chinese would have cut the meat into small portions beforehand and use chopsticks to eat, instead of using their teeth while implicitly holding on with their hands.
But the Book of Rites has one, revealing mention of chopsticks:
If the stew [羹 geng] be made with vegetables, chopsticks should be used; but not if there be no vegetables.
Traditional Confucian commentators explain this by saying that a stew with vegetables is any stew with solid ingredients, since even a meat stew would use vegetables to spice up the taste. But a stew without vegetables is just a clear broth. For the former, you should use chopsticks to pick out the solid ingredients; for the latter, there's no choice but to use a spoon or drink from the bowl, since it's all liquid.
The use of chopsticks to eat solids in a stew, of course, would be a natural evolution of the use of chopsticks to stir those ingredients in the stew beforehand.
So according to the etiquette of early China, there were three main ways to eat: the hand, for grain and meat; the spoon, for soup; the chopstick, for things in the soup.
But beginning in the late Zhou, things were beginning to change. First, the elite began to cut meat and other large foodstuffs into small portions, which was both quicker to cook and a more refined way to eat than tearing off with your teeth. This is evident from the culinary habits of Confucius:
Confucius did not dislike his meat to be finely diced. [Analects 10.7]
This made it possible to eat meat with chopsticks instead of your hands. And indeed, by the 140s BC, a Chinese emperor could shock and humiliate a general by "honoring" him with a huge chunk of meat, undiced and with no chopsticks to eat it with. Said official looked in vain for chopsticks, showing that the use of chopsticks for meat was established by this time.
Somewhat later, rice and wheat began to supplant millet as China's dominant grain, and steamed rice and flour-based foods like noodles and dumplings are both much easier to handle with chopsticks than steamed millet is.
By the third century AD, eating with your hands had fallen entirely out of fashion in China, replaced with spoons and chopsticks. This continues in Korea, where spoons and chopsticks are complementary, but in China proper even spoons were largely supplanted by chopsticks by the fourteenth century due to a variety of cultural factors.
What was the role of forks in all this? Chinese forks, like early chopsticks, were cooking utensils. The Shang forks you're thinking of were not, as a rule, used to eat food. And unlike chopsticks, which made their debut in the dining room because they were useful when picking out solids in hot stews without scalding your hand, there was no real situation where forks were more useful than fingers or chopsticks. And when the development of diced food allowed the use of chopsticks to expand, since forks weren't already established as a popular dining utensil, they couldn't take advantage of the situation to become as widespread as they would be in the West. And later on, since forks and knives were associated with the culinary traditions of steppe nomads like Jurchens and Mongols, there would have been stronger cultural reasons not to adopt them.
This answer largely draws on Q. Edward Wang's Chopsticks: A Cultural and Culinary History.