Mason here.

There's not a clear-cut answer to this question. If you want a ELI5 answer, most people will go with the year 1717 as a founding date- but things are a bit more detailed than that.

Here are the basics:

• In 1717, the UGLE (United Grand Lodge of England) was founded. Today, legitimate Masonic lodges derive their official status from being formally recognized by UGLE. This helps you avoid joining a lodge that's trying to scam you, or one that's injecting their own non-Masonic teachings into Masonry.

• Everything today is structured in a "Grand Lodge" model, where each state in the US has its own Grand Lodge that independently carries out administrative duties within that state, each in turn being recognized by UGLE. There is no national overseeing Masonic body, and no worldwide Masonic body. UGLE recognition is a matter of legitimacy rather than hierarchy.

• While 1717 was the start date for the modern Grand Lodge model (as a result of four lodges in England getting together to found the UGLE), it's self-evident that those four lodges (as well as additional lodges) existed prior to 1717, and were seeking to standardize what they'd already been practicing for a long time.

• There are solid records going back into the 1600s of Masonic lodges in England acting in a manner that look like what we consider Freemasonry to be today, with some minor differences.

Once you start getting into the early 1600s and before, no one has a definitive claim on what was going on. That being said, plenty of people write books speculating about what was happening during this time. Many people try and tie Freemasonry to the Knights Templar, but these connections lack sufficient evidence to stand up to scrutiny. They're not impossible, just not supported sufficiently.

There's also the question of when things transitioned from "operative Masonry" to "speculative Masonry" (essentially when Lodges transitioned from being solely guilds of actual stonemasons with their own rituals into Lodges that allowed individuals from other professions that modeled rituals after stonemason traditions). There's not a solid answer for when this happened, but it would be fair to say that it was some time during the Middle Ages.

At this point, we can get into the question of manuscripts. The earliest of these is the Halliwell Manuscript, also known as the Regius Poem from 1390. Having read through the translation of it, I can say that there are some very clear but subtle phrases used that leave me personally without a doubt that Freemasonry, in some form, existed at this point in time. Admittedly, it was likely a distant cousin of what we now know. This manuscript makes a claim about Masonry being brought to England in the 900s AD, but the trail really goes cold at this point.

TLDR; 1717 is an easy date to reference, lots going on in the 1600s, scattered details from 1300-1600 that are fun to speculate on, and insufficient evidence before that.

I'm not a historian, but I am a Freemason who is interested in the Fraternity's history.

Let me first outline what we know to be true:

1. Guilds of stoneworkers existed throughout the history of Europe.
2. Starting in the 15th century and possibly earlier in what are now England and Scotland, we begin to see fragments of documents which indicate the presence of a more intellectual strain within these guilds which has similarities to modern Freemasonry.
3. By the 17th century, recognizable Lodges of "Freemasons" were present in both countries
4. In 1717, several English Lodges banded together to form the United Grand Lodge of England, giving Freemasonry its modern form of governance (individual, autonomous Grand Lodges).
5. From 1717 to the late 18th century, Grand Lodges were formed all over the world, including Ireland, Scotland France and the Americas at first. These Grand Lodges maintained their ties to the original ideals of Freemasonry through a system of mutual recognition, not centralized control.

Okay, so now let me cover the more speculative elements.

Freemasonry is a philosophical extension of many strains of influence ranging from stonemasonry (obviously) to the early schools of the famous Greek geometers to the Abrahamic faiths to the chivalric institutions of Europe. There are echos of all of these influences within Freemasonry, but it isn't any one of them. It was the interaction between the increasingly influential stonemasons and their highly secretive guilds and the aristocracy which brought about this strange hybridization and brought about the modern Lodge.

References:

1. Masonic manuscripts on Wikipedia collects many of the earliest sources of Masonic history and links to a number of useful references for them.
2. ULGE's History of Freemasonry
3. MSANA's History of Freemasonry
4. Grand Lodge of British Columbia and Yukon's history index