The cases Anthony cites suggest that elites spread their language when they create a social group that outsiders want to and can join, within which their language is spoken. For the Turks in Turkey, religion was a big factor, and the people who ended up speaking Turkish were the ones who converted to Islam. The ones who remained Christian still spoke Greek. In Algeria and Tunisia, the French offered people disaffected with the traditional Arab-speaking world a cosmopolitan alternative, and the result is that many intellectuals and nonconformists speak French. The point is that people made a conscious choice to join the invaders' world, and that included taking up their language.
So how did those mostly male warriors impose their Indo-European languages on so many people?
The simplest answer, it seems to me, is that they recruited people to join their social order, and that across Europe people wanted to join it. They brought with them, not just languages, but a way of living that was hugely appealing to many people.
One important point about Indo-European warbands is that they could serve as a means of recruitment. We know that some Anglo-Saxon kings had Welsh noblemen among their retinues. Indo-European invaders may have done the same thing, welcoming the sons of Europe's older elites into their households. After all, their sisters were already there ahead of them. A noble's household was conceived of as a sort of school where highborn young people learned the ways of their ancestors, and they often included foster children sent to that school by the noble's leading followers. This household could therefore become a mechanism for spreading culture and language.
Perhaps their religion was also appealing. We still love stories of the Greek, Norse, and Sanskrit gods, and the basic rituals they practiced remained powerful for thousands of years. The invaders' religion certainly mixed with that of the lands they conquered, picking up new gods and transplanting cosmic events into local caves. But the myths of Indo-European speakers across Eurasia have enough in common that we know they date to before the great migrations, which is remarkable fact in itself.
That world -- the world of the Iliad and the Odyssey, of the Táin Bó Cúailnge and the other Irish sagas, of the ancient Indian Vedas -- still fascinates us today. Perhaps it also fascinated the people of Europe and northern India 4,000 years ago. That fascination, plus ambition, may have led some of them to join their conquerors. The genetic data seems to tell us that the Indo-European elite was at first composed mostly of immigrants, but that the percentage of Steppes DNA declined over time as they slowly merged with Europe's older inhabitants. We know they kept key parts of their own culture, especially their languages, so that means other people were joining them.