Because it was the only major culture there. Japan is an island and Korea is a peninsula. So on one side you have China and on all other sides you have vast ocean. In terms of being influenced, there wasn't any other options.

The cultures of China, Japan, and Korea were in close contact for millennia before they had any real interaction with the West. That's plenty of time for significant cultural exchange, even if the process is extremely slow. In fact, the conventional wisdom is that East Asian countries (especially Korea and Japan) have Westernized shockingly quickly given that the bulk of that process has happened over just the last century.

Religions in particular can spread very quickly before then sticking very tightly. Buddhism is primarily practiced in East Asia but started in South Asia. Islam started in Arabia but has adherents from Nigeria to Indonesia. Christianity started in Jerusalem (kind of) and is now practiced by at least a substantial minority on every continent. By comparison, taking Confucianism across some mountains or a sea is pretty easy.

Well, why was Greek and Latin culture so influential throughout all of Europe? I think the most compelling explanation for both of these is that they were literary cultures in the Ancient period. For generations Japanese scholars were learning Chinese to study the ancient classics, all of which were written in Chinese long before Japanese writing had even been invented. And they didn't start using Chinese characters and loanwords necessarily because they just loved Chinese culture so much - it was because all the ancient and prestigious texts used those words and characters, the same as we write with the Roman alphabet and use plenty of ancient Greek loanwords in English.

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.

>From the little I know about East Asian history, Japan and Korea took forever to embrace Western culture

It took around 50-60 years for Japan. To be fair, I start counting from people showing up with enough military capabilities to force treaty upon the government. (Perry's expedition)

It's easy to spread culture when you have a lot of people. But those people all need to be fed. China has one of the most fertile farming regions in the world. Many of the rivers from the Himalayan mountains deliver nutrients east through China, providing great farmlands. It's why China currently has the world's largest population. On the other side of the Himalayas, India enjoys the same benefits which is why it has the second largest population.

The other East Asian nations simply could not produce as much food. Korea is on a mountainous peninsula in the colder north. Japan is an island nation. Southeast Asia is a tropical jungle.

Korea and Southeast Asia "embraced" Chinese culture when threatened with China's large military. Japan took elements of Chinese culture through exchange and trade but it didn't stop China from trying to invade.

China has done the best at having culture, society, and governance for an extended period of time. China had ships ten times larger than anything in the west before the time of Columbus. They could have colonized the world long before anyone else but never desired to. They kept records well over an extended period of time so could build off the advancement of individuals. It was easy for developing cultures around them to copy their work instead of duplicating it.

Edit: I get it CPP bad but that's just from 70's to today. China has a much longer history.