Details like this are known because they are given in some source.

This particular example is recorded by General James Wilkinson in his memoirs. He gives the story in a footnote, recalling it as probably told him by General Shelby, and writes that he was told the names of the brothers but does not remember them. See page 115 in volume 1 of his memoirs: https://archive.org/details/memoirsofmyownti01wilk

Draper, in his history of the Battle of King's Mountain, King's Mountain and its heroes, quotes Wilkinson, and continues "It is not improbable that these two brothers who confronted and killed each other, as related by Colonel Shelby, were of the Goforth family, of Rutherford County, North Carolina. At least, four brothers—Preston Goforth on the Whig side, and John Goforth and two others in the Tory ranks—all participated in the battle, and all were killed. It was a remarkable fatality." This is the version that makes the story widely-known today. See pg 314 in Draper: https://archive.org/details/cu31924032752846

As is usual for this type of event, it is reported by an eyewitness. Eyewitness accounts are given in, or are the sources for, letters, newspaper articles, memoirs, medal citations, and so on. Inclusion in a widely-read history of the event can make the particular story well-known.

Such stories are not always reliable. Eyewitnesses can be mistaken. The reports are often passed on by others, who might forget details, conflate multiple accounts, embellish the stories, etc. Further, it is often the unusual and remarkable events that are passed on as such stories.

For modern histories of recent events, interviews with participants are often an important source, and such histories are often very rich in this kind of detail.