Why did the writings of Confucius become influential in China?
The most common reason given for this is the fact that China was in such a state of upheaval that people came to value Confucius's ideas.
Confucius's ideas were centered around how to maintain harmony in a society. He emphasized, for example, that all people needed to understand what their place in society was and needed to act in ways that were appropriate to that place. If they did this, society would be stable and harmonious.
As you can imagine, this was the sort of philosophy that would go over well when things are chaotic in a given society. If you are (as China was) in a period where there are all sorts of powerful people leading armies to fight one another, devastating the land and killing people, you would want a philosophy that would promote stability. You would want to have some way of making all the chaos end.
So, Confucius's ideas came to be influential because they emphasized stability and harmony at a time when those things were lacking in China.