Buddhism was the first wide-scale missionary religion. It helps something's popularity that monks went all over the place to teach it.

This is a bit like asking how did Christianity influence the development of Europe.

There's literally hundreds of books I can recommend on this topic in general but it sounds like you're just asking for general histories of these Asian countries since Buddhism was present almost at any reasonable definition of their "beginning," of course with the pretty big exception of China, and even that feels like a loaded statement.

For starters, I'd recommend:

China: A History by John Keay

Buddhism in Chinese Society: An Economic History from the Fifth to the Tenth Centuries by Jacques Gernet (dry as bones but good)

Buddhism and Buddhists in China by Lewis Hodus (old but a classic)

The Introduction... and The Assimilation of Buddhism in Korea ed. by Lewis R. Lancaster. (again, dry but excellent)

Korean Philosophy: Its Tradition and Modern Transformation ed. by Korean National Commission for UNESCO (not a historical text, but you get to see the progression of Korean thought, including of Buddhism, through time)

And while there are plenty of Japanese books on Buddhism, I haven't read very many I'd feel comfortable recommending as sources with the exception of Zen at War by Brian Victoria, who also has an article (it's been a while, but I think it's just a shortened version of Zen at War) in Buddhist Warfare which is an excellent anthology (ed. by Michael K. Jerryson) covering the history of Buddhism's relationship with violence and conflict in a variety of settings (including in original source material, China, Japan, SE Asia, and Tibet).

The "Why" is a much much harder question to answer and we could be here for years discussing it as the answer will be different for why it was so popular (though note that Buddhists were at every extreme from being genocided to ruling as Emperors from its entry into East Asia to the present day) from region to region.

EDIT: Formatting