I'm also interested in which groups it developed in particularly, how widespread and influential they were, and how common the belief is today. I've seen some talking about it on the internet, but I don't know how common it actually it is.
This article from the Detroit Baptist Theological Seminary may be helpful.
Essentially, the KJV was translated from the Textus Receptus, or Received Text, which was Erasmus's Greek Translation, if I recall correctly and was derived somewhat from Erasmus's own Latin translation, which drew from the Vulgate. The Vulgate was the first Latin Bible which was translated from what was to become the Greek New Testament and the Septuagint, the Greek Old Testament. The Vulgate did not rely on the Hebrew Old Testament, rather the Greek Translation, for many different aspects.
The Luther Bible, Tyndale Bible, the KJV, and others are all translated from the Textus Receptus. The KJV wasn't the first English translation of the Bible, but it was endorsed by the King, and the finest scholars in the Britain began work on it. It's been criticized for preserving certain tropes of the day, such as the Diving Right of Kings, as well as using very high-brow language and poetic phrasing. The Koine Greek of the original text was very much layman's tongue, except perhaps Luke's Gospel.
As scholarship grew and older manuscripts were found, a Revised Version was published. These older manuscripts corrected scribble errors and shed light on vague passages. For instance, the last several verses of Mark are now considered to be scribal additions due to continued scholarship.
A man named John William Burgon began writing about the new Revised Version. Though many modern KJV-only advocates believe that the Textus Receptus is perfect and inherrant, Burgon wrote, "Once for all, we request it may be clearly understood that we do not, by any means, claim perfection for the Received text. We entertain no extravagant notions on this subject. Again and again we shall have occasion to point out that the Textus Receptus needs correction” (Revision Revised, p. 21, fn. 2). Burgon believed that it was important to maintain the text that had been preserved by the Bishops and the church for decades. He believed that since the Textus Receptus had been preserved for so long, it must be the correct text, so Burgon rejected the Revised Version.
Bill Combs, of the Detriot Baptist Theological Seminary, writes, "The King James-only movement believes that only the KJV is the Word of God. All other English versions are corrupt since no modern Bible version (except the New King James Version) is translated from the Textus Receptus (TR) Greek NT, which is considered to be without error."
The movement spread to American with the beginnings of fundamentalism. Combs writes, "The name fundamentalism was not coined until 1920 by Curtis Lee Laws, but the founding documents of the movement can be traced to a series of 12 volumes produced between 1910 and 1915 titled 'The Fundamentals: A Testimony To The Truth.' The 90 essays quote from the KJV but also from the RV (or its American edition)." In 1924, Phillip Mauro write a book entitled, Which Version, Authorized or Revised? This criticized the thousands of changes from the KJV, or the Authorized Edition, to the Revised, claiming they were completed without authority.
Through the 20th century, various fundamentalist preachers pushed for KJV only. Combs writes, "Even the well-known fundamentalist leader John R. Rice initially promoted the RSV in his widely-read paper, Sword of the Lord. From then on a stream of KJV-only literature began to appear: Jasper James Ray, God Wrote Only One Bible (1955); Edward F. Hills, The King James Version Defended (1956); David Otis Fuller, Which Bible? (1970); Peter S. Ruckman, The Christian’s Handbook of Manuscript Evidence (1970); etc. Though the RSV was popular in mainline churches, it was not widely accepted in evangelicalism, and not at all in fundamentalism."
Basically, the idea began in England as sort of a counter-movement to tradition. Remember, the KJV is a Protestant translation endorsed for years by the Anglican church. It spread across the pond and is carried on by some to this day.
How prevalent is it today? Combs says that the KJV is not widely accepted anymore by mainline churches. Independent churches, like most Baptist churches, make their own decisions regarding doctrine. Some churches will post signs outside that say 1611 KJV Only or something to that effect. Some takes it to the extreme. You can find several videos of sermons where the preacher proclaims that the Jesus himself carried the 1611 KJV. Such extreme positions aren't common, though someone people like the KJV due to tradition. Even in memorizing Scripture today, children will be taught "For God so loved the world that he gave is only begotten son that whosoever shall believeth in him shall not perish but have ever lasting life," which is from the KJV.
As Combs notes, the New International Version published in the 1980s accounts for more than 40% of Bible sales.
Sources:
http://www.dbts.edu/2012/04/03/the-modern-kjv-only-movement/
http://www.dbts.edu/2012/03/21/dean-burgon-and-the-revised-version/
http://www.dbts.edu/2012/03/28/the-kjv-only-movement-comes-to-america/
You might be interested in the /r/AcademicBiblical subreddit, which might be able to answer your question.