Why did Buddhism become so influential during the Age of Disunity in China?
The "Age of Disunity" refers to the period roughly 220-589 C.E. when China was split into several different kingdoms and dynasties. Throughout this period Buddhism was developing in China. We can point to a few factors in this growing popularity.
The first was simply the expansion of Buddhism beyond India. Most of this expansion was along the Silk Road through Central Asia. Figures such as Bodhidharma entered China and began spreading a Buddhist message. Bodhidharma founded the Chan (Zen) lineage in China; his dates are in dispute, but he was likely to have lived in the 5th century C.E. The first Buddhist texts were only just translated into Chinese in the second century C.E. Another prominent Mahayana Buddhist missionary to China in this period was Kumarajiva (344-413 C.E.). many missionaries traveled the Silk Road into China, and this road was sometimes traveled in the other direction. Faxian (337-422 C.E.) was a Chinese Buddhist who traveled to India to learn more about Buddhism.
There may be philosophical reasons for Buddhism's popularity as well. Confucian philosophy deals with the organization and harmonizing of society, with moral cultivation to better oneself and one's relationships. Yet we can see in this period that society was in turmoil. Buddhism may also have appealed to those attracted to its message that suffering lies at the heart of human experience. To put it another way, seeing the failure of the Confucian ideology to hold society together may have interested people in a philosophy that tells people that joy comes from being detached to the troubles of the world.