Thanks for all of the great responses, yo!
Much of the information we have about the Indo-Europeans is extrapolated from linguistic data. Similarities in languages in disparate geographical regions led linguists and historians to conclude they came from a common source(There are some, like Chomsky, who would argue grammar and form is inherent to language and the common threads in culture's languages belies, not a common source, but a common biological/neurological structure between humans. I have no dog in the fight, but it is an interesting argument nonetheless.EDIT: This claim is unsubstantiated and reflects my poor understanding of Chomsky.) The archeological evidence itself is scant and subject to conjecture.
Keeping that limitation in mind, some historians have argued the Indo-Europeans were more technologically advanced than adjacent societies,
We can see that the presence of words for pottery, domestic animals and agriculture in the Proto-Indo-European lexicon argues that the community was atleast Neolithic and that it would be senseless to assign the Proto-Indo-Europeans to the earlier hunter-gather societies of the Mesolithic. More, we encounter in Proto-Indo-European vocabulary not only words that one associates with the original 'Neolithic Revolution' but also the lexical residue of what Andrew Sherratt has termed a later 'Secondary Products Revolution.' These include the secondary uses and products of domestic animals which can be seen in dairy products, wool and textiles, wheeled vehicles, yokes, ploughs and the domestic horse. Sheratt suggests that these secondary products only emerged in Europe several thousand years after the initial appearance of the Neolithic economy. (Mallory, 1989)
From later on in the same piece:
The economy which we envisage from the linguistic evidence for the Proto-Indo-Europeans is well-enough mirrored in the archaeological evidence. Agriculture is attested, but nowhere does it seem to have been the primary subsistence component of the economy. Rather, the emphasis of the archaeological remains appears to be primarily on stockbreeding...
...The most significant linguistic-archaeological correlation among the domestic animals is the horse, which is known at least from the fourth millenium BC. It is not simply present, but all the evidence indicates intense exploitation and role in ritual.
The horse, in particular, has been considered by many historians to be a game-changer with regards to warfare. It allows armies to move quickly, establish effective supplies chains and present the appearance of large numbers. This, in my opinion, would probably be the main reason for the Indo-Europeans' large scale success.
Source: In Search of the Indo-Europeans by J.P. Mallory
Don't forget the Middle-East! The Persians and Medians were another IE group which grew to prominence and supplanted the previous ruling class. The original theory on the spread of PIE languages across so wide an area was that the Indo-Europeans were just that good at killing their enemies and taking their lands. But I gather that this view has fallen out of favor with modern linguists and historians. I would also love to hear from someone who specializes in this field on what the current consensus is.
Here is a recent article on the PIE homeland that touches upon this subject a bit: David W. Anthony and Don Ringe, "The Indo-European Homeland from Linguistic and Archaeological Perspectives"
[p. 209-210] "A later spread from the steppes requires, for Europe as well as Asia, a more complicated sociolinguistic model of language shift. Long-established agricultural communities must have given up their non-IE languages and shifted to IE languages in the absence of empire or conquest. Military conquest is not a viable vector of language shift in the Bronze Age, because there were no empires and no administrative structures to effectively extract wealth from distant territories among the competing armed chiefdoms of Bronze Age Europe. The sociopolitical context would have been small-scale tribes and clans, mobile in some places and settled in others, arranged in shifting alliances between regional chiefs and armed with weapons displayed prominently in graves and in ritual hoards. During the Bronze Age, after 3300 BCE, in the Pontic-Caspian steppes, and during the Late Neolithic, after 2800 BCE, in much of Central and Western Europe, status associated weapons became more common in Yamnaya, Usatovo, Corded Ware, and Bell Beaker graves, and warfare was both glorified and institutionalized with the emergence of formal warrior identities in many regions... Archaeological evidence alone does not show how the kaleidoscope of competing European chiefdoms conceived of political power and clientship, which are highly relevant to language shift. But information about political power and clientship is contained in the PIE vocabulary... as well as in early Indo-European poetics... elements of which have already been used in discussions of Bronze Age politics..."
The authors then go on to describe how the IE institutions of the roaming warrior band and "patron-sponsored public feasts and guest–host relationships" helped spread IE culture and language to the detriment of pre-IE cultures and languages, concluding:
[p. 214] "All these institutions could have provided the framework for recruiting outsiders into IE-speaking societies in social positions offering vertical mobility, horizontal reciprocity, and the possibility of immortality through poetry, made more attractive by generosity at feasts. The threat of violence from war-bands and the possibility of restitution for damages could have added some compulsion. A new, more productive, and mobile herding economy and long-distance trade in metals and other valuables (salt, in some regions) probably brought prestige and power to IE-speaking societies after 3300 BCE."
TLDR: a combination of possessing superior technology (new metallurgy techniques, new types of fast, wheeled transport) and new social institutions that both encouraged and coerced non-IE peoples that IE peoples settled among to abandon their old ways in favor of IE culture.
For what it's worth, Steven Pinker writes in The Language Instinct that what we know about them by reconstructing their vocabulary points to either a steppe culture called the Kurgans being the first to use horses in warfare and sweeping across Eurasia (kind of like the mongols) OR an Anatolian culture taking up agriculture and slowly spreading in search of new arable land.
Follow up question: Does the evidence actually point to Indo-European languages supplanting other languages or merely "interbreeding" with them? (And thereby providing the common material that unites all the resulting hybrid languages.) That would change the question to be not why Indo-Europeans were good at outcompeting, but why were they so nomadic?