First off, answering why a culture is the way it is is always a bit... iffy, at best. Especially cultures that are this old and have no written records, or even written descriptions from other cultures. Old Europe is especially tricky, because we don't even have a reconstructed language and daughter cultures to pull from like we do with the steppe cultures (if one is using the Proto-Indo-European steppe origin theory). Also, be careful assuming a default culture, which your question flirts with. You could just as easily ask why Old Europe was peaceful and egalitarian.

Second, my understanding of the hypothesis is that the "peaceful egalitarian" label is only meant to apply to the people of Old Europe (which refers to a specific culturo-geographic area within Europe, not simply all of Europe at that time), and that the people in other parts of Europe and outside of Europe aren't covered by the hypothesis.

Third, the "steppe nomad warriors" in this age should probably not be understood as something like the modern stereotype of steppe nomads in the vein of the Mongols or Huns. Warfare that came as part of movement from the steppes into Old Europe would more likely have been tribal warfare in the form of raids and things like that. Don't think of it as a coordinated invasion with big armies.

Fourth, the precise nature of the cultures of Old Europe is debated, especially when it comes to precisely how peaceful they were. In particular, the idea sometimes presented of a pacifist utopia is... overstated at the very, very best.

Fifth, what exactly caused the collapse of Old Europe is debated. The steppe tribes are there, but so are possibilities of climate change, environmental degradation, conflict over declining resources. And of course a combination of factors is a possibility as well.

Now, with the caveats and background done: David W. Anthony, in The Horse, the Wheel, and Language, partly attributes an increase in warfare among/by the steppe tribes to the development of herding economies on the steppes. These are economies where one can become directly wealthier by owning more animals, helping a more pronounced social hierarchy to develop. The use of easily transportable (because they can walk themselves) animals as wealth allow for increased conflict in the form of cattle raids and the like, which can easily evolve to revenge raids and endemic conflict.

Anthony also argues that the adoption of horseback riding helped, because it allows people to travel faster and farther, and also makes herding easier. This allowed cattle raids to become longer and bigger, and also for animal herds themselves to become larger, leading to richer chiefs and more resources (and reasons) for warfare.

Anthony also (very) tentatively partly attributes the patrilineal nature of the steppe cultures to cattle herding (which he says often leads to brothers staying together and to the development of hierarchies based on warfare and power). He cites a case where the adoption of cattle raising among Bantu-speaking tribes in Africa apparently led to a transition from matrilineal to patrilineal kinship systems. But again, this is a tentative idea presented as an aside. It's not a main point of the book or even the chapter or section and it's treated as such, meaning it's not discussed in any real detail and shouldn't be taken as an actual argued hypothesis (at least from Anthony's book, it's possible others have covered it in more detail).