"How" is that it likely comes from the term "Mob Boss". Bruce Lee may have played a part in popularizing this term, with movies like "Big Boss" and "Game of Death" (1971/1972). Also, Coppola's "The Godfather" came out in 1972 as well; and in it several times Michael/Don Vito are referred to as "boss" or "your boss". It's also a popular trope in other media like comic books... Kingpin, a mob boss, can be traced back to 1967, and The Penguin can be traced back to 1941. So the idea of big-bad leaders of villainous organizations being "bosses" was established in etymology well before video games came out.

Then, you have video games. It was definitely adopted pretty early on, with both the Kung-fu and original Metroid (NES) user manuals directly using the term (Metroid only says mini boss, but that kinda alludes that the main big bad is THE "boss") in 1985/1986 respectively. However, this term wasn't necessarily universally adopted; in British game journalism you can see games like Double-Dragon (1987) having their bosses referred to as "super baddies" or games like Outzone having "big guardians". But that's not to say no one called them "bosses" at the time, just that the term wasn't necessarily universal. Nintendo was clearly an early adopter of the term in their own works, but it was likely already being used colloquially. In fact, the 1988 english console adaptation of Double Dragon does directly refer to Jimmy as the "Shadow Boss", and calls the last area the "Hideout of the Boss".

Galaga arcade sets in 1981 called the big enemy at the end of some levels the boss as well; calling them "Red Boss" and "Blue Boss". But it's not clear if this was just their name, as the "boss" of the other aliens, or a reference to the modern "boss" term. Either way, it's either the earliest example of a video game company calling something a boss themselves, or it is a likely precursor of the term. More importantly, Joystik magazine refers to it as "The Boss" in 1983, and that could have had a huge impact on the etymology considering how popular the game was, and how easy it could be for readers to conflate "biggest baddest enemy=boss" in future games, no matter what Namco meant by it.

Last, we have the 1974 Dungeons and Dragons on Plato. Given that contemporary games at the time were pong/pac man/asteroids, it's likely that this was THE first inclusion of a "boss fight" in video games, with the golden dragon. While I don't know of any media that directly refers to it as a "boss", it may be the earliest example of the trope in video games.

And what this brings us to is, D&D; and it's precursor, Chainmail. Chainmail was created in 1971, and D&D in 1974 (which, as an aside, it's amazing that a text-based video game was created in the same year, given the niche audience at the time; it really shows how people were affected by this style of storytelling and gameplay). Unlike contemporary video games, these were primarily story driven with over-arching plots and dungeon crawls, while video games were usually reaction/tactics based (like the aforementioned pong/pac man/asteroids). They frequently used "boss" type enemies by comparison, even in their official campaigns. However, I haven't been able to find any specific usage of the term "boss" in older D&D booklets... Now I'm invested so I'll continue to look and update this post if I find any. But my point here is that it's possible D&D players were already using the term before the concept hit video games, and given the overlap in early-adopters/timeline between the two media, it could be the original source.