A lot of it has to do with history, apparently. I was reading Baseball in Japan on Wikipedia, and I came across this very interesting historical tidbit:

In 1891, Ichiko challenged the "whites only" Yokohama Athletic Club to a match-up on the diamond, only to have the request refused, as the Yokohama squad refused to play against non-Caucasian players. As a result, the team from the Christian missionary school Meiji Gakuin offered to play Ichiko and subsequently handed them a decisive defeat. Humiliated, Ichiko began developing an intense training philosophy wherein players would train to the point of complete physical exhaustion for the improvement of the team. This training ideology would serve as the foundation of the Japanese game well into the 20th century. In 1896, the Yokohama Athletic Club (fielding a team composed mainly of sailors) finally agreed to play against Ichiko and were defeated 29 to 4. It was the first recorded international baseball game in Asia.

Baseball was introduced to Japan by Hiroshi Hiraoka, an engineering student who was exposed to baseball while studying in the United States. Hiraoka then introduced the game to his co-workers at the Japanese National Railways (now JR Group), and later formed Japan's first baseball team. However, baseball did not become prominent until the formation of the Ichiko (First Higher School of Tokyo, now a part of the University of Tokyo) baseball team, and after they won in 1896, baseball really took off in Japan.

In Korea, baseball was introduced there by American Christian missionaries, and as Korea fell under the Japanese Empire (where baseball took off), it became popular as well both there and in Taiwan.

To put the long story short, basically baseball became popular in Japan because the Japanese took the effort to make it popular, and from there it spread to Korea and Taiwan.