Didn't I understood the text or is the 'why' not really part of it? I expected more than a vague 'because it slightly existed and then hands are free to do things and brains got bigger'. I miss the point.

They don't discuss a "why", so much as present data on the "how" and "when". If this work is valid and reliable, then it will be up to later research to propose and test hypotheses as the why.

In a nutshell, the paper basically says that the lateralization that led to the predominance of right-handedness occurred around the time humans became bipedal and around the time of neuroanatomical expansion, possibly related to bipedalism.

In other words, before these two changes, we used all four limbs for locomotion and had no preference for either forelimb for grasping. Then one or two things happened and right-handedness predominated. It seems that that neuroanatomical expansion took over the areas of the brain that previously allowed our left hands to be as capable as our right hands.

I write "one or two things happened" because it wasn't clear to me from the paper whether the neuroanatomical expansion that led to lateralization was necessary to and part of bipedalism, i.e., caused by our locomotion bits taking over other parts of the brain to manage our balance, or whether it was merely coincident with it.

Interesting questions asked and answered, more research needed.

Actual study here: https://journals.plos.org/plosbiology/article?id=10.1371/jou...

My take is that when they added extra factors to the Bayesian model, the plot was such that humans were no longer outliers.

Whether or not that's scientifically rigorous, or even interesting, I leave to others to determine.