It comes from Protestants wanting to discard tradition and councils and claim the Bible is the one source of doctrinal truth. As most "Bible Christians" in the US started with the King James Version, that has been the ultimate source of truth for them and most of their theology is based on it. New versions of the Bible that don't line up with their arbitrary "gold standard" are a threat to the foundations of their faith as a result, and have to be regarded as wrong.
Note that in most King James Only articles and websites, the reasons listed for why the other translations are wrong have nothing to do with textual or historical accuracy. They're entirely based on how the other translations do not support their belief system.
Example: One word used in the Old Testament is translated "sodomites" in the KJV, and as such there are verses like "He drove the sodomites out of the land." It is now recognized that the word actually meant "shrine prostitutes." However, when KJV purists see the new reading of "He drove the shrine prostitutes out of the land," their reaction is to declare the newer version wrong and blame the Homosexual Agenda because the Bible isn't supporting their bigotry as much anymore.