China has been the dominant force in the region for around 2000 years. For long stretches of that time many countries had formal suzerainty treaties with China, sometimes even direct vassalage.

Chinese characters were in use in Korea, Japan and Vietnam for centuries before anyone they were exposed to anything Western.

Confucianism is much more of a moral philosophy than a religion. It doesn't focus on origin myths, the afterlife, gods or spirits. It's a set of philosophies around the relationships of people. Much closer to Plato's Republic than the Bible.

Confucius lived over 2500 years ago. Jesus' teachings made it all over the world in less time than that. Indonesia is pretty far from Mecca and it's become the largest Muslim country in the world just 1300 years after Muhammad. It's not at all surprising to see significant cultural diffusion to nearby neighbors after all that time.

And people move around too. Over thousands of years there are many factors that make people pick up their stuff and check out the neighbors; wars, natural disasters, famines, rumors of new opportunities. China has had a very large population for a long time. There have been a lot of Chinese people moving to neighboring countries. So to a large extent Chinese culture extended throughout Asia because so many Chinese people wandered around Asia.