I've seen this claim asserted by numerous Vedic/Yogic gurus, and of course Hindu nationalists. While the ancient traditions that influenced what we know today as Hinduism are probably very old indeed and go back far into prehistory, the Sanskrit language itself, from a linguistic standpoint is certainly not the oldest, nor did it originate in India. Both linguistic and genetic evidence strongly suggests that the language has its origins in a geographic area ranging from Eastern Europe to Central Asia -- i.e. the Eurasian steppes.
Sankskrit is an Indo-European (IE) language belonging to the Indo-Iranian branch, which includes all the Indo-Aryan languages derived from Sanskrit, in addition to the various Iranian languages like Persian, Kurdish, Pashto, Ossetian, and many others. Old Vedic Sanskrit and Avestan (the language of the oldest Zoroastrian scriptures) are so similar that it can be argued that they can be considered two dialects of the same ancient language; this language itself is probably a form of proto-Indo-Iranian, which in turn is an offshoot of the language the Corded Ware Culture of Eastern Europe spoke about 4,500 years ago. The Western branch of the Corded Ware, the language of those who remained in Eastern Europe and migrated further West, was probably the ancestral tongue of the Balto-Slavic branch of the IE tree. Indeed if we compare Sanskrit to other IE languages we find the greatest degree of lexical similarities with the Baltic and Slavic languages of today.

Long story short, the peoples who brought Indo-Iranian languages to Central and South Asia were from the steppes, probably nomadic horseback-riding warrior tribes. These quite-mobile migrants conquered many lands and assimilated the locals of whatever places they brought under their dominion. There is much genetic evidence to back up this hypothesis. The primary male Y-haplogroup of Eastern Europe is R1a. And indeed in Indo-Iranian speaking populations we see a high frequency of R1a among males. In Northern India we find R1a somewhat common among men of Brahminical lineages, and less common among those of other modern Indian populations, yet still there. And in modern European populations we find a lack of native South Asian genetic markets, which suggests that an "out of India" theory for IE language dispersal is completely baseless and little more than a weak, emotions-based assertion made in service of modern Indian identity politics. Ultimately ALL form of identity politics adopt the habit of making absurd and baseless historical claims. Identity politics is all about ego, as opposed to spiritual enlightenment.
Finally, linguists can generally pinpoint the region of origin of a language family by noting which region has the highest diversity of branches of that language family. Distinct branches show very old splitting within the family. For example, the Afro-Asiatic (AA) family (which contains the Semitic, Berber, Cushitic and old Egyptian languages) sees its highest branch diversity in Northeast Africa (around the Horn region and Ethiophia), suggesting the perhaps Afro-Asiatic speech began there and then later spread to other regions. The in the Middle East, all AA speech is uniformly of the Semitic branch, suggesting a single population brought proto-Semitic into the Middle East from somewhere else at some point in prehistory. Likewise, with Indo-European we see the greatest branch diversity in Southeastern Europe (which might suggest this are and not the steppes as the oldest area of IE speech), not Central or South Asia. In SE Europe we see Greek, Albanian, Thracian (now extinct), Slavic, and very close by in Anatolia we now have Armenian and we once saw Anatolian (Hittite and related tongues) and Phrygian branches. And then of course in the interior of Europe we see different branches like Romance (derived from classical Latin), Germanic, Celtic, Baltic, Slavic, ect. In Asia all we really see are the Indo-Iranian and now-extinct Tocharian branches. In other words, Asia seems to have been the receiver rather than giver of IE languages. None of these facts constitute any sort of Western or European supremacy; they're simply facts which archaeology, linguistics, population genetics and history can prove many times over.
As a final note: For those beholden to the idea that India is the font of all the world's spiritual wisdom and greatness (there's a slight hint of truth to this), the acceptance that outsider populations brought the Sanskrit language in into the Indian subcontinent is an inconvenient theory to entertain. I'll get more into why that is at a later time.
Sankskrit is an Indo-European (IE) language belonging to the Indo-Iranian branch, which includes all the Indo-Aryan languages derived from Sanskrit, in addition to the various Iranian languages like Persian, Kurdish, Pashto, Ossetian, and many others. Old Vedic Sanskrit and Avestan (the language of the oldest Zoroastrian scriptures) are so similar that it can be argued that they can be considered two dialects of the same ancient language; this language itself is probably a form of proto-Indo-Iranian, which in turn is an offshoot of the language the Corded Ware Culture of Eastern Europe spoke about 4,500 years ago. The Western branch of the Corded Ware, the language of those who remained in Eastern Europe and migrated further West, was probably the ancestral tongue of the Balto-Slavic branch of the IE tree. Indeed if we compare Sanskrit to other IE languages we find the greatest degree of lexical similarities with the Baltic and Slavic languages of today.

Long story short, the peoples who brought Indo-Iranian languages to Central and South Asia were from the steppes, probably nomadic horseback-riding warrior tribes. These quite-mobile migrants conquered many lands and assimilated the locals of whatever places they brought under their dominion. There is much genetic evidence to back up this hypothesis. The primary male Y-haplogroup of Eastern Europe is R1a. And indeed in Indo-Iranian speaking populations we see a high frequency of R1a among males. In Northern India we find R1a somewhat common among men of Brahminical lineages, and less common among those of other modern Indian populations, yet still there. And in modern European populations we find a lack of native South Asian genetic markets, which suggests that an "out of India" theory for IE language dispersal is completely baseless and little more than a weak, emotions-based assertion made in service of modern Indian identity politics. Ultimately ALL form of identity politics adopt the habit of making absurd and baseless historical claims. Identity politics is all about ego, as opposed to spiritual enlightenment.
Finally, linguists can generally pinpoint the region of origin of a language family by noting which region has the highest diversity of branches of that language family. Distinct branches show very old splitting within the family. For example, the Afro-Asiatic (AA) family (which contains the Semitic, Berber, Cushitic and old Egyptian languages) sees its highest branch diversity in Northeast Africa (around the Horn region and Ethiophia), suggesting the perhaps Afro-Asiatic speech began there and then later spread to other regions. The in the Middle East, all AA speech is uniformly of the Semitic branch, suggesting a single population brought proto-Semitic into the Middle East from somewhere else at some point in prehistory. Likewise, with Indo-European we see the greatest branch diversity in Southeastern Europe (which might suggest this are and not the steppes as the oldest area of IE speech), not Central or South Asia. In SE Europe we see Greek, Albanian, Thracian (now extinct), Slavic, and very close by in Anatolia we now have Armenian and we once saw Anatolian (Hittite and related tongues) and Phrygian branches. And then of course in the interior of Europe we see different branches like Romance (derived from classical Latin), Germanic, Celtic, Baltic, Slavic, ect. In Asia all we really see are the Indo-Iranian and now-extinct Tocharian branches. In other words, Asia seems to have been the receiver rather than giver of IE languages. None of these facts constitute any sort of Western or European supremacy; they're simply facts which archaeology, linguistics, population genetics and history can prove many times over.
As a final note: For those beholden to the idea that India is the font of all the world's spiritual wisdom and greatness (there's a slight hint of truth to this), the acceptance that outsider populations brought the Sanskrit language in into the Indian subcontinent is an inconvenient theory to entertain. I'll get more into why that is at a later time.