There is a band of related languages from western Europe to India which have similarities (in grammar and vocabulary) which can only be explained by descent from a common ancestor, or at least from a group of similar dialects, in the remote past. The Indo-European languages of India and those of Iran are particularly closely related. Unless the language family began in India and spread outward from there, which seems highly unlikely, then someone had to have brought it to India, whether Aryans or Berlitzers.
"So how did the Indo-European languages show up in India? Did Berlitz send a team of instructors and teach the proto-Sanskrit language to millions of people?"
I don't know, and neither do you. They might have originated on the subcontinent and moved outward via trade etc.
THere is no archaelogical evidence of a mass invasion in that timeframe, that I know of.
There are those who consider the Aryans as indigenous to India. That there is a connection between Indo-European languages is not being contested. Simply these people have the opinion that proto-Indo-European spread from India, rather than from Central Asia (the main hypothesized origin area for the original Indo-Europeans) or southwestern Asia (Mesopotamia after Babel when one of the splinter groups had the ancestor language of the Indo-European languages--in a Creationist model).
Why do the French speak a language related to Latin, and not a Celtic tongue? Why do the Irish speak English, and not Gaelic? How did the Egyptians stop speaking Coptic and start speaking Arabic? How did the Anatolians first move to speaking Greek, and then to Turkish? What about the Prussians moving from speaking a Baltic tongue to German?
The Indians taking up a language related to the nearby Iranians, Tocharians, and Afghans is hardly a revolutionary process in the history of the world never seen before. Its not as though the Iranians/Afghans never ruled the Indus Valley and north Indian plain.
"Indo-European languages show up in India?"
It went to Europe from India and not vice versa. The oldest languages in the family of Indo-European languages are to be found in India not Europe.