I didn't read thoroughly, but it looks like they were studying twins who had been raised in the same home, and finding big correlations of personality attributes. (“Participants were volunteer twin pairs [...] eligible if they were 16 years old or over and raised together in the same home.”)
How does that prove that genetics (as opposed to identical age, same family, similar relationship to the family, close contact with each-other, etc., not to mention similar interpretations of the meanings of questions on a personality survey due to shared environment) was causal?
You sure didn't read thoroughly; the next sentence after that one explains it.
They were comparing monozygotic and dizygotic twins, which allowed them to measure to the extent that they were similar because of being raised in the same family at the same time. Comparing monozygotic twins raised in different families is another way to measure hertibility. Both have their issues, of course. But measures of the heritbility of personality generally do hover around .5 _+/- .1 in other studies.
How does that prove that genetics (as opposed to identical age, same family, similar relationship to the family, close contact with each-other, etc., not to mention similar interpretations of the meanings of questions on a personality survey due to shared environment) was causal?