In AD&D, there were dual-wielding rules in 1e (1977) but they were not specifically identified with rangers. A high Dexterity did mitigate the associated penalties, however, meaning that characters otherwise concentrating on a high Dexterity, like rangers and rogues, were natural fits for the style. In 2e (1989), rangers got the class ability to avoid the usual -2/-4 dual wielding penalties.
Aragorn, one of the major fictional inspirations for the ranger, did double wield at least once in the Lord of the Rings (at Weathertop), but it wasn't a major part of his deal. Many other historical figures dual wielded including Davy Crockett and other pioneers that commonly fought Indian style with tomahawk and knife, which has some clear ranger analogues. I remember being inspired re: dual wielding by Madmartigan in Willow (1988) but that's too late to drive the 2e rules.
Much of the shift seems to be game mechanic driven. Not being as armored or as hard-hitting as the fighter, the ranger needed something to boost them mechanically, and dual wielding was chosen due to the alignment with high-Dex.
But What About The Drizzle?
Those who weren't around at the time seem to think that this ranger ability stemmed from the fictional exploits of Drizzt Do'Urden in several Forgotten Realms novels. Drizzt the famous double wielding drow ranger was published in 1988 after being made up on the spot in 1987, really too late to affect the course of the 2e rules. Besides, in 1e drow could (debatably in some circles) double wield; his ability to do so is this more about his drow-itude than being a ranger. Not that he was much of a ranger either, his character sheet on wizards.com says Ftr10/Bbn1/Rgr5.
David "Zeb" Cook, designer of 2e, said when asked where the two-weapon ranger came from,
I'm not sure where the ranger took shape, though I know it wasn't an
imposition because of Drizzt. (Frankly, I've never read more than bits
of the Drizzt series.) It was more to make them distinct and it fit
with the style and image.
There's an episode of the 1970s Kolchak: Night Stalker series that includes a shapeshifter called a rakshasa. It also involves a vulnerability to blessed crossbow bolts/piercing weapons, which isn't part of Hindu tradition (I think Ravana, greatest of the rakshasas, was actually killed by a round thrown weapon called an asthra). Here's a summary of the episode.
Gygax confirmed the inspiration in a forum Q&A post in 2005:
I was a fan of Kolchack, the Night Stalker, when it first aired, and sure enough they had a rakshasa as a monstrous evil on that show. I liked the idea of the demon being destroyed by a blessed wooden crossbow bolt, that being akin to the stake through a vampire's heart, so I went with that in the MM.
Nowadays I'd be less prone to allowing so easy an answer to the threat of a rakshasa, although not many adventuring parties are equipped with a crossbow and blessed bolts.
Cheers,
Gary
Best Answer
The inspiration for it, like for many other things, comes from Tolkien and The Lord of the Rings, specifically the part where Aragorn, Legolas and Gimli chase down the orcs that have taken Merry and Pippin. Here's the direct quote:
— J.R.R. Tolkien, The Lord of the Rings, Book III, ch.2, "The Riders of Rohan"
I would guess that this, as well as some other quotes from the same chapter (like "As before Legolas was first afoot, if indeed he had ever slept", and other supporting quotes) are the inspiration for "elves don't need sleep" trope that D&D embraced