The earliest form of spoken Sanskrit used in the Rigveda is typically dated to about 1500 BCE, and is without any doubt an Indo-European language, in this case part of the Indo-Aryan subgroup.

Talking about linguistic history that old can get a bit complicated. Linguists have spent the last 250 years or so working out predictable patterns in how language changes to estimate how old any form of that language is. Rigvedic Sanskrit is a particularly interesting case for that because the Rigveda does not appear to have been written down in any recognizable form before around 500 BCE. Between 1500-500, it was passed on as an oral tradition, but remained linguistically very similar to the original form even as the day-to-day language around that tradition changed. It would be comparable to medieval English speakers passing on Beowulf in proper Old English, while everyone hearing the story was actually using Shakespearean English in day to day life.

The Aryan migration into India was one endpoint of the broader history of the Indo-European language family. Around 4000-2500 BCE, a language we call Proto-Indo-European was spoken in the area of modern Ukraine. This language was never written down, but so many known languages are derived from that original language that we can reconstruct a lot of its history. Based on the words shared across many languages, scholars can identify the area where it started, and a rough sequence of when different groups of migrations left the original Indo-European group.

One of the last migrations out was a large eastern movement identified as the "Indo-Iranian" group, around 2500 BCE. By about 2000 BCE, the Indo-Iranians had settled in a large swath of Central Asia, and based on the oldest known languages in this category (Sanskrit and Avestan) we know that they called themselves Arya, which is conventionally rendered as "Aryan" in translation. Around 1700-1500 another migration pushed south into what is now Iran and Afghanistan. The segment of that migration that went toward Afghanistan continued on into India developed into the "Indo-Aryan" or "Indic" language group, starting with early Sanskrit and then diversifying from there.

The group that entered Iran in that same migration actually kept going south and west, with some of them eventually reaching modern Syria, where they became the ruling class of a kingdom called Mittani. We know that the Mittani were part of this Indo-Iranian migration because a few segments of prayers, names, deities, and descriptions of horses are nearly identical to the same words and phrases used in the Rigveda. By all indication, this language was constrained to just the upper class, and over time fell out of use entirely aside from a few select words.

The Indo-Iranians that remained in Central Asia were the linguistic ancestors of the whole Iranian language family, but only really started spreading past the northern fringe of modern Iran around 1000 BCE.

Back in India, most linguists think that the early Sanskrit speakers probably entered a territory where most people spoke Dravidian languages, similar to those that still dominate southern India. Linguistic historians point to small pockets of Dravidian languages still spoken in the north today. As a general rule, small linguistic minorities rarely move somewhere new and carve out an area where they remain dominant. More often than not, a small territory with a minority language is the remnant of a language or languages that were once more common.

Aryan migration which was claimed to bring about the current Indian culture and the sacred Indian texts such as the Rigveda.

This is true n the sense that the Indo-Aryan languages and elements of the culture described in the Rigveda have had great influence on Indian culture as a whole. However, much of current Indian culture, and almost all of the later Vedas and other works of Indian epic and literature were composed in India itself (or Pakistan in the sense that it was part of ancient India). The Indo-Aryan people didn't just arrive and set up their existing culture in a new place. It adapted and incorporated elements of local belief and tradition.