The way the male scientists report this is 'Some men mated with lots of women', which is one way of putting it. Another way of putting it is that for an extended period in prehistory men were twice as likely as women to die without reproducing themselves. And when you consider how dangerous it must have been for women to give birth in a cave, this means that men must have been dying at a rate we don't see in any population these days: more than 50% of men must have died without any offspring at all. If you think that more that two thirds of the men and women alive in the Middle Ages have offspring alive today (despite the black death etc.) then you can see that there must have been a male-skewed die-off quite unlike anything in recorded history.
It's possible there was some terrible disease or parasite which affected men only - those immune have left offspring. Or perhaps there was some male-only occupation (hunting or war perhaps) which was way, way more dangerous than the most dangerous hunting or war we know of. So perhaps the tribes consisted largely of women, with a small number of adult men, many of them severely injured or sick.
Or perhaps there were lots of adult men, but a few of them exercised such iron fist control over the rest that most men never had sex. There is nothing like this in any human society today, except a small number of extreme religious cults. It's hard for me to imagine how this could be enforced in a hunter-gatherer society, I would imagine that young men and women would just run off. But perhaps it happened.
Whatever the reason, daughters would be twice as likely as sons to pass on DNA, so in genetic terms they were a much more secure 'bet'. Food and other resources may therefore have been directed more at girls, and this may have resulted in a feedback loop - girls are more likely to survive, so they are given more resources, so they survive etc. Thus the small natural survival advantage of girl babies over male may have been amplified by child rearing practice.
So, there are many possible ways that we could have ended up with this strange pattern of variance. I think all we can conclude is that ancient society was quite unlike anything that exists nowadays. A society of scarce and damaged men, or a society of male warrior-virgins, or of pampered daughters, nobody knows. But 'men mated with lots of women' doesn't really capture the level of strangeness these findings imply.
(of course it's possible the whole study is founded on incorrect notions of DNA creep and repair, or their model is way off, but I'm taking it here at face value)