When and why did the modern literal 6-Day creation theory become popular? Historically, Creationists tended to fall into two broad categories, those who subscribe to the Day-Age theory (the idea that the days in Genesis 1 represent ages of indeterminate length, not 24 hour periods) or the Gap theory (the idea that between Genesis 1:1 and 1:2 was an indeterminate gap, but the days were 24-hour periods). 

At least up until WWII, these were the predominant Creationists views, with the idea of a young earth being vehemently attacked even by some Fundamentalists and Creationists.


  Advocates of [creation science] read the first chapters of Genesis in a way that allows for no life on earth before Eden and no death before the fall.
  Until the last few decades most creationists would have regarded such notions as unnecessarily extreme. By the late nineteenth century even the most conservative Christian apologists readily conceded that the Bible allowed for an ancient earth and pre-Edenic life. With few exceptions, they accommodated the findings of historical geology either by interpreting the days of Genesis 1 to represent vast ages in the history of the earth or by separating a creation "in the beginning" from a much later Edenic creation in six literal days.
  -- Introduction to The Creationists, 2nd Edition, 2006, by Ronald L Numbers, p.7


Jump to 2016, and the theory of a literal 6-day creation has become very popular, and is certainly the most talked-about of the creationist theories. In comparison, the Gap and Day-Age theories get much less attention, even being considered heresy (or nearly so) by some literal 6-Day creationists.

What accounts for this change in attitude, and when did it occur?
First of all, the dispute has definitely been around for a long time. Here is a link defending an allegorical day interpretation. It references Origen (not a saint) and St. Augustine:

http://biologos.org/common-questions/biblical-interpretation/early-interpretations-of-genesis

St. Augustine specifically makes the point that there was no sun or earth on the first day, so how could days be counted?

And here is a link using the writings of St. Basil to defend a literal day period for creation.

http://creation.com/genesis-means-what-it-says-basil-ad-329379

The debate clearly existed in the church pretty early on, but it was never accepted as grounds to accuse someone of being a heretic, no councils of the church mention the dispute is what I mean (that I know of).

As for our current state, I think Ross Douthat's book Bad Religion does a good job of addressing this. So I'll sort of paraphrase him and hopefully not oversimplify too much.

In chapter 1 Mr. Douthat points to 2 major movements within Protestant Christianity at the beginning of the 20th century: modernists/accomodationists who want to change the church's doctrines to be more palatable to skeptics and fundamentalists who want to preserve the church's beliefs. Modernists were willing to challenge the inspiration of the scriptures, virgin birth, reality of the miracles of Christ, and even the resurrection and incarnation. Fundamentalists insisted on preserving all these doctrines and that you had to believe these these things to be Christian. The picture given is that some, or most, fundamentalists became overzealous in their defense and strayed into an error of their own by insisting on literal interpretations simply to oppose whatever modern theory they felt threatened them and condemn Christians who entertained them.

He specifically points to the 1920's as when the terms "Evangelical" and "Fundamental" became associated with strict biblical literalism. This decade is when the biblical literalists in Tennessee took the issue to court and attempted to ban evolution from school textbooks and lost.

However, even orthodox Protestants weren't anywhere near uniformly "fundamentalist" in this sense. Some more specific information comes in chapter 4 of the book. In the 40's Fuller Theological Seminary required its students and faculty to affirm that the Bible was as "exact a guide to geology and biology as it was to the history of salvation". However in the 60's the position of most of the faculty and students shifted to limited inerrancy, meaning that the Bible was only inerrant in the realm of theology. 2 professors even published a book with the thesis that the rigid understanding of inerrancy was invented in the 17th century. This triggered a dispute which resulted in the "International Council on Biblical Inerrancy" which defended a strict position, but with allowances. J.I. Packer (supporter of strict inerrancy) is quoted at the council as saying:


  I believe in the inerrancy of Scripture ... but exegetically I cannot see that anything Scripture says, in the first chapters of Genesis or elsewhere, bears on the biological theory of evolution one way or another ... Scripture was given to reveal God, not to address scientific issues in scientific terms


So Mr. Douthat concludes that even though inerrancy was upheld in the 70's its implications were reinterpreted to allow for non-literal understandings and truths.

I suppose that his view can be summarized as fundamentalism rising to answer (badly) the challenges of modern scientific skepticism and liberalism. He even labels fundamentalists as anti-intellectual in some cases.

I wouldn't really point to the 20th century as when fundamentalism kicked off though; remember Galileo was condemned because his views didn't fit with certain literal interpretations that the church had at the time. I would view this literal fundamentalism as more of something that comes in waves, just as liberalism does. The lifetimes of Galileo and Darwin are two big waves that often get talked about in history. The sexual revolution of the 70's was another one.

I would say the currently the literal trend seems to be subsiding. Just the other day my Evangelical friend (who listens to way too many podcasts) noted how allegorical interpretations are becoming more popular. You could even point to Rob Bell as an example of an Evangelical pastor who has some very allegorical (and probably heretical) interpretations.

A real answer to this question probably involves a whole big study of 2000 years of Christianity. I don't necessarily think taking creation as a literal 6 days is bad, but I think that Genesis has some much deeper spiritual meaning that will be missed if it is simply used as a beating stick for more liberal Christians.