How it is possible that Sanskrit is related with Greek and Latin?

Aryan is in origin the same word as Iran which the Shah of Persia adopted after a visit to Hitler, who used "Aryan" (incorrectly) as a synonym of Indo-Germanisch (German for Indo-European). I began my own study of historical linguistics from a book I found in a second-hand shop. This contrasted the simple vowel system of Sanskrit with that of other Indo-European languages and claimed this showed the "purity" of Sanskrit: the vowels of all the others being distortions of their primitive form. Unfortunately my book had been published in 1870. Years later the discovery of Hittite (spoken in Asia Minor) proved the contrary: that the original common Indo-European ancestor language had had a quite complicated vowel system and that its simplification in Sanskrit was the most obvious result of how that language had been influenced by the phonetic habits of the early Indians who had spoken Dravidian languages before adopting the Sanskrit of the Aryan invaders. Apart from that however, Sanskrit, Hittite, and the primitive Greek spoken on Bronze-Age Crete are the earliest documented Indo-European languages and our chief source to understanding what the original late Stone Age common Indo-European must have been like. Compared to these languages of the third millenium BC, Latin is not documented before c.400 BC, Gothic (the earliest known Germanic language) before 300 AD, English not before 600 AD, and Old (liturgical) Bulgarian (the earliest documented Slav language) not before 700 AD. But just as an example of how rapidly they all changed, even primitive Greek had already turned every initial S into an H. On the other hand Hittite watar is closer to its English and Dutch cognate water than to the Classical Greek hydros.