So we know the Proto-Indo-Iranians migrated down from Russia into Iran and India, and then migrated into Europe.
What I'd like to know is what made their language so special as to overtake the indigenous linguistic forms of the populations of Europe and Asia (if there were any to begin with!)?
Do we know anything about the pre-"Aryan" world and the languages people spoke?
This has been a question long in my mind, and this is probably the only place I can think of that offers a sliver of a chance in answering this.
You could also try on /r/linguistics, as you might find some info from PIE scholars about how the Indo-Iranian languages spread into Iran/India. I will however answer this question as best as I can. I must stress though that, although I have read 'The Horse, the Wheel and Language' (which I suggest you read too), I am not a terribly qualified linguist or historian.
Essentially, the Proto-Indo-European peoples occupied the Pontic-Caspian steppes around 3500-3000BC, which ranged from what is now Ukraine to the Urals. These were mainly pastoral herders of cattle and horses, and may or may not have been semi-nomadic, who towed their belongings around on wheeled carts and chariots. Over time, there were migrations of people towards Europe that split off from the Proto-Indo-Europeans, and who spoke languages ancestral to the Celtic/Germanic/Italic etc.
The Indo-Iranian homeland, however, was in what is now Central Asia. These Indo-Iranians would have remained semi-nomadic, and would have herded across vast expanses of territory. Tribes that descended from the Indo-Iranians include Aryans and modern Iranians/Kurds etc, but in fact Indo-Iranian tribes once covered most of Central Asia as well as the steppes of Eastern Europe (eg. the Sarmatians, Scythians etc.). The ones in Central Asia, however, were assmiliated by Turkic tribes around 2000 years ago.
The Steppe regions of Central Asia had much harsher climates than those of, say, India or Western Europe, hence the propensity of pastoral herder tribes with less sedentary lifestyles than those in agriculturally fertile areas. They would have therefore been spread over very vast areas, as the modern ethnic groups of Central Asia are today.