I don't necessarily agree that Times New Roman is the "most accepted" and "most well known" font, but it's gotten around plenty since 1991, and this has to do with the history of TrueType fonts.

TrueType wasn't the first outline font (Type 1 fonts, made by Adobe and used in Postscript, I believe came first), but it became part of Mac OS 7, and Apple licensed it freely for use with Microsoft Windows. Previously, Mac OS and Windows used bitmap fonts for all parts of the interface.

Microsoft released Windows version 3.1 in 1991, which came with a slew of features which were game-changing, like OLE, the Registry, common dialogs, and TrueType fonts. On OS 7, Apple released several TrueType fonts, among which were included Times Roman, Helvetica, and Courier. Windows included three fonts by default to match these, in terms of style, functionality, and metrics: Times New Roman, Arial, and Courier New.

From there after, anywhere Windows went, along with it came Times New Roman, and it was the default font for Microsoft Word documents as well. During the 90s, Windows came to dominate on desktop computers; Microsoft Office became the standard for documents; and in the late 90s, Internet Explorer began to dominate the web, using whatever fonts came along on the system to display web pages.

So it was an accident of history (the massive popularity of TrueType and Windows) that led to Times New Roman becoming massively popular.

I'll have to look through my books to properly source, but I'm sure I learned most of the details about what font features were introduced when from Petzold's Programming Windows series of books.

Times New Roman was designed in 1929 for the Times of London by Stanley Morrison of the Monotype corporation, one of the two big type foundries 100 years ago. The other biggie was Linotype and they licensed it, calling their version "Times Roman". Alternatively, Adobe writes that the the Times' commissioning of the font was in 1931 and that Morrison was only the head of Monotype, not the font's designer. Adobe also makes claims about the origins and basis which are argued or refuted by pretty much everyone else.

I don't have enough source material to argue the merits of these points (I can't call the Times since it's a holiday weekend) and they're unimportant for answering the questions. At some time before WWII the font was designed by Big Name in Typefaces and was licensed to The Other Big Name in Typefaces.

As technology progressed, so too did the reach of this basic font. The name of your standard serif body type depended on the technology you were using. CompuGraphic? "Times Roman". Mergenthaler? "Times New Roman".

TNR was designed for use in blocks of text. Legibility was the key concern. Second to that came compactness; TNR is somewhat narrower than other serif body fonts, but its ubiquity (and utility origins) have made it distasteful for most people in typography. Garamond is similar but more expensive -- Adobe makes a lot of money from that one.

So, to answer the original questions:

How did Times New Roman become the go-to,

Because it was available from both the big type houses 100 years ago and remained easily licensed.

most accepted,

Because for most of that time, few people had a choice over typeface used. For most people, the typeface they used was determined by the manufaturer of the typewriter they bought: Underwood, IBM, Olivetti, etc.

and most well known font?

That would be Helvetica, thanks to the documentary and book. It's a default, one easily overridden these days.

TNR is common because it's the default. Because it's cheap and easy to get. It's a workhorse and will get the job done.

Source: I used to set type, cold, line, and electronic (Linotype, Mergenthaler, CompuGraphic [77xx & MCx series], Wang, IBM, Lexitron, PCs since 1977. I have some printed references if something is in question, but it may be in storage.

To understand Times New Roman, we have to start with William Starling Burgess.

From here:

But typographer Mike Parker discovered that in the early 1900s—before Times New Roman or Plantin existed—Boston yacht builder William Starling Burgess drew samples of a new font and sent them to Monotype’s U.S. affiliate. Burgess lost interest in the project, but his drawings were never returned. Parker theorizes that years later, Burgess’s drawings were passed along to Stanley Morison, who secretly used them as the basis of Times New Roman.

Like much of typographic history…it's murky. Saying Morrison/Lardent stole TNR is clickbaity, but saying it was entirely his design is not really true, either. At least that's my takeaway.

We gotta remember: type has to look really similar for it to be readable. It changes at a glacial pace in comparison to other artforms. Saying a typeface is exactly like another typeface is hard to do in retrospect, because it could have just been a fashion in the air, and the world was ready for TNR or something exactly like it. Or it could have been the work of Burgess and it was totally ripped off by Morrison.

It's hard to say because not everyone working in type gave that much of a shit.

Type was a very blue collar job during this period. If you find old photos of printers from this time, you'll see a bunch of missing fingers — they would take the solvents and rub them around their nose and lips to get high, then run the machines and lose bits of their body. This was the environment that type-related jobs existed in. That level of ruffian-ness probably did not exist so much at Monotype — I'm thinking a bit more Wild West America with that image — but it's important to think of this context, of type as a blue collar art form. If the guys working then were still alive, they really may not have remembered, it was just a job. And Burgess seemed to not really care if/when he became aware: he was interested in yacht design, clearly a much less pedestrian form of design.

To answer the question:

Microsoft Word made the statement in this question true. It cemented it with the status you see.

Its use in newspapers and various technologies that brought typesetting to the masses throughout the 20th century lead up to its obvious inclusion in Windows.

It also has something to do with the name: it suggests long-form reading in a newspaper, it's New, and it comes from the Romans — every school kid in western society has learned that the Greeks and Romans created the basis of western society, and they've probably been shown a picture of Trajan's column.

Gerard Unger, a very important name in type design, did a PhD a couple years ago on Romanesque letterforms. You can buy Alverata, his typeface based off of these forms. The interesting thing about the Romanesque is 1) they were spread from Scandinavia to North Africa, from Spain to the Byzantine empire, 2) The Roman capitals are noticeably not spread like that, they're in Rome, and 3) they're quite different from the Roman capitals.

So like a lot of things, there's a story implied by the name Times New Roman that feeds into ideas of being a continuation of a great empire…but of course recent scholarship has punched some holes in that.

EDIT: Just to add a follow-up thought, something I think is worth considering with this question: what font, or fonts, built the framework for TNR's rise? TNR is sort of this little branch of a large evolutionary tree, and really the interesting crux points in that evolution happened much earlier. If there's any Dutch/French/Belgian people interested in this thread, a visit to the Plantin-Moretus Museum is a must. It's the trade and influences shared between the low countries and England that give rise to this sort of type.

A note to commenters: This isn't really the appropriate forum to share your favourite fonts. Please consider having that discussion elsewhere. Discussion here should be limited to the history behind OP's question.

Others have already mentioned the most obvious cause of its fame: that both Apple and Microsoft licensed it for ubiquitous use in software, in 1984 and 1992 respectively.

Butterick's Practical Typography has a section about Times New Roman that tries to trace its design history before the age of PC word-processors, starting from its commissioned design for the Times of London in 1929 by Stanley Morison of Monotype. There are a few competing theories about the previous influences that inspired TNR.