Of India you mean? There's no disputing that the Indo-Aryan languages originated outside of South Asia and spread to the subcontinent from the northwest at some point before recorded history. Exactly when and where those languages came from taps into the wider debate on Indo-European origins, which is always a provocative topic, but I think at this point we can be reasonably sure that they originated in West-Central Asia and were brought to India via the Iranian plateau sometime in the Late Bronze Age. Genetic evidence suggests that the spread of the language also involved a fairly substantial movement of people.

Now, the details of how that linguistic/demographic shift took place is a lot more debatable and I don't think it fits anything you could call an "Aryan invasion theory". The concept of the Aryans as a prehistoric ethnic group actually originates in India with the Rig Veda, probably the most valuable source on how the Indo-European languages actually spread. However more recent readings of the Rig Veda characterise the arya as a social stratum rather than an ethnicity. They were set apart from the dasa by their nobility and superior religious practices, and notably not because of their language or ethnic descent. This obviously makes the idea that the Aryans were an external group that brought a language into South Asia with them less persuasive. Even if we do equate arya identity with speakers of Indo-Aryan, if it's a prescribed social status rather than a transmitted ethnic affiliation then it's entirely possible that the indigenous population adopted it and became 'Aryans', rather than being replaced by them.

The "invasion" part is even more problematic, as this evokes the idea of a cataclysmic military campaign by the Aryans that wiped out the preceding, indigenous Indus Valley Civilisation (IVC). This narrative is typical of early twentieth century explanations of prehistoric cultural change that are now generally seen as hopelessly simplistic and lacking any basis in fact. There is no material evidence of an Iranian culture steamrolling its way across northern India. Nor is there evidence of large-scale violence, battles, cities being burnt to the ground and the like. There isn't even anything in particular that says the IVC was "wiped out" by anyone – urbanism declined for sure, but there's a lot of continuity in the material culture of the IVC and post-IVC periods.

If I were to proffer a synthesis of what really did happen (and I should stress this is heavy on speculation), it would be that what we're seeing with the 'Aryanisation' of South Asia is similar to the process of Indo-Europeanisation in Europe. In both regions the populous but relatively egalitarian agricultural societies that flourished in the preceding period seem to have hit a sort of malaise at the end of the 3rd millennium BCE. This was perhaps related to a combination of deteriorating climatic conditions, social stresses following millennia of population growth, and the inherent limitations of Neolithic farming economies.

These conditions seemed to favour more mobile, pastoral existence, and a group of nomadic herders with distant origins in the North Pontic steppe took great advantage of this and spread widely across Eurasia. Their way of life was quite different to that of agriculturalists like the IVC: livestock was a source of tangible economic and symbolic wealth held by individuals, leading to the emergence of significant social hierarchies; their use of horses and chariots gave them an edge in raiding or small-scale warfare, and may have allowed to them set themselves up as patrons of sedentary farming villages; and their mobility gave them social connections across a wide area, probably another source of social prestige.

There would be no need for these nomads to violently conquer the existing agricultural societies of South Asia. They simply moved in and pastured in marginal areas with little value to farmers; maybe even grazed herds on their fields after the harvest. But you had a situation where the old order was stagnating, people were struggling to make a living the way they used to, and yet here was a group of newcomers with large, prosperous herds; intriguing new ideas about individual wealth and status; military superiority; prestigious goods from exotic locations; and a language that could be understood not just in your own village but across most of the known world. It stands to reason that the existing population would be tempted to adopt the language and customs of these newcomers, which I'd suggest was the crucible of the arya identity that crops up later in the Rig Veda. Over the centuries this prestige-driven cultural assimilation led to the complete replacement of the preceding (presumably Dravidic) socio-linguistic order with a new, Indo-Aryan-speaking one across much of Northern India.

(This is essentially a slight elaboration of David Anthony's theory in The Horse, the Wheel, and Language, which I'd highly recommend as further reading.)