I've written on "Aryanism" and how it came to be associated with Whiteness, and traces up to the early 20t-century and Nazism, so will repost the older answer below:

How did "Aryan" come to mean blonde haired blue eyed white supremacists when the Aryan people were part of the ancient Vedic culture, who by geography would not have been white or blonde?

I would preface of course first to say that trying to find logic in racial pseudoscience is often an exercise in futility. Thinkers almost always contradict in some way as they each are, in the end, creating their own proofs for an untruth, and even single theories often have obvious points of contradiction and inconsistency. But in any case, that dispensed with, the term "Aryan" as we think of it in the strains of racial pseudoscience where it is most commonly associated with Nazism is most significantly a result of the writings of Arthur de Gobineau, a Frenchman with pretensions of aristocracy (self-styled as Comte de Gobineau) writing in the mid-1800s, and best known for our purposes for this work Essai sur l’Inégalité des Races Humaines.

Gobineau wasn't doing anything particularly new in his approach to writing on ideas of race and his work reflected common ideas of a noble Germanic race which was superior to all, terms the "Nordic" myth, and which traces back at least another century to the writings of another Frenchman, Henri de Boulainvilliers, and which subdivided the people of France into the Nordic, Alpine, and Mediterranean types, placing the Nordics at the top, and exemplifying them as, put succinctly, the "descendants of ancient Germanic tribes, the originators of all civilization, and the only peoples capable of leadership".

Gobineau took this and expanded on it, but writing in the 1850s, added onto it new ideas which had entered into the intellectual milieu of the time, and one of these was the use of the term "Aryan", which although first applied in India, was a linguistic term created by a British official to describe the ur-language shared by a vast swathe of the globe, what we would now term Indo-European. The idea of Aryanness language was promoted by several figures before Gobineau, but he made it essentially his own with the way he melded it to racial thinking. The logic of course was quite simple, if the Nordics were the originators of all civilization, and Aryan language the roots of all language, presumably the Aryans and the Nordics were the same thing. But of course he doesn't stop there either, adding in ideas about purity of the blood and corruption caused by mixing of the races.

Whites were the superior race, Aryans the best of the whites, and mixing with the inferior races - "White", "Yellow", "Black", which he based on the three sons of Noah - risked the very existence of civilization itself. The purer the Aryan blood, the better the civilization, the weaker, the worse. Aryans had propagated far and wide - settling Europe, Iran, and India - hence why traces were found even in the latter, but even in Europe, where the Germanics represented the purest specimen remaining, they faced great dangers of decline which Gobineau already believed to be well advanced and needed to be arrested quickly. Gobineau was hardly ignorant of the place of India in all of this, and in fact based much of his work on readings of the Rig Veda which he believed supported the story of Genesis. Figueria summarizes other Sanskrit literature he drew on thusly, using them as evidence for the eventual degradation of the Aryans in india due to dilution from long intermingling with the "aborigines":

The Mahabharata bore witness to the manner in which Indian society had been invaded by foreign elements.9 Savage vices, absent from the Ramayana, appear full-blown in the history of the Pandavas, who had been raised to divine status in order to veil the blood sins of their mothers. In other words, Gobineau read the epics as chronicles of non-Aryan promiscuity and Aryan battles to avoid the dilution of their bloodlines.

This he held up as a warning for what Germany was going through, and the threat to Aryan purity of the Germanics, which they needed to work to stop.

Later thinkers built off this further. Ernst Haeckel's The Riddle of the Universe: At the Close of the Nineteenth Century would prove to be an especially influential one in his specific focus on tying in the Aryan Myth with Germanic nationalism and ideas of Volk, and the preeminence of the blonde, blue-eyed ideal, which of course are so closely entwined with the popular image of Nazi ideas which would come later.

But perhaps none is a more important bridge between Gobineau and the Nazis than Houston Stewart Chamberlain, who wrote Foundations of the Nineteenth Century. An Englishman by birth, he was fascinated by Germanness, marrying Wagner's daughter, and writing German propaganda through the First World War. Foundations pushed the Aryan Myth heavily, and was a work admired by Hitler himself, and of many things, is known for being a key part of the attempt to demonstrate that Jesus was, in fact, not Jewish but of Aryan persuasion, which of course ties into the virulent anti-Semitism pushed by all of the writers mentioned here, Chamberlain being a key part in developing Hitler's belief that the Jews needed to be entirely removed from German society.

It also of course it worth mentioning the American Madison Grant who wrote The Passing of the Great Race: or The Racial Basis of European History which likewise played in important part in the propagation of the Aryan Myth, and played a key role in the creation of the eugenics movement in early 20th c. America.

But in any case, that is the rough summary of it all. "Aryan" was a term latched onto by 19th century thinkers for their writings on racial pseudoscience as it fit well with their ideas of a master race, a ruling class, which had once been spread far and wide. Gobineau, as did others, used esoteric and wildly ahistorical readings of texts and evidence to support this view, explain away incongruities, and place their Germanic ideal in the place of this master race in their way of thinking. Although India was a key part of the origin of the term "Aryan", these racial thinkers were careful to explain why India itself, of course, was not the exemper of the Aryan race, and instead held it up as an example of how far the white race could fall.

Sources

Evans, Richard J.. The Coming of the Third Reich. Penguin, 2005.

Figueira, Dorothy Matilda. Aryans, Jews, Brahmins: Theorizing Authority through Myths of Identity. State University of New York Press, 2003.

Saini, Angela. Superior: The Return of Race Science. Beacon Press, 2019.

Sussman, Robert Wald. The Myth of Race: The Troubling Persistence of an Unscientific Idea. Harvard University Press, 2014.

Tattersall, Ian & Rob DeSalle. Race?: Debunking a Scientific Myth. Texas A&M University Press, Sep 2011.

/u/jan_van_bergen has previously written about Why did Hitler specifically choose to call his desired race "Aryans" rather than "Nordic" or "Germanic"?

/u/killfile has previously answered Why was being Aryan such a big deal to the Germans?