Well, they don't.
Whenever people go through the records and count stuff up, they don't happen near the full moon more than at any other time. It's an artifact of how memory works -- two things that coincide are more likely to be remembered than each on its own. Indeed, as a rule, people's statistical impressions are frequently wildly inaccurate.
It's not something that gets better as you go back in time. Preachers fumed about the May festivities and the young men and women, but a researcher went and tallied up the baptisms of illegitimate babies, and found that there wasn't a bump nine months after -- though there was one nine month after August. Perhaps the harvest was more convenient for frolics -- or perhaps it was the warmer weather.
Similarly, there was a culture in Africa that would assure the anthropologists that virtually all accidents were caused by witchcraft, and virtually all witches were women. He tallied up the accusations and found that about half of the actual accident prompted charges, and men were in fact more likely to be accused.
Which means that, if you go and read some primary source, and find that someone asserts that this, that or the other is more common than another -- you might be wise to be very wary indeed about it.
You get a nice example of that in Christian polemic in the Roman Empire. Some Christians wrote as if virtually all the babies abandoned died, but others, as if they were all taken up by brothel owners. This was probably not a regional difference in the babies' fates, because it turned on the issue. If the writer was refuting charges that their secret rites involved sacrificing a baby, they would indignantly point out the way the pagans killed their own; for charges of sexual immorality, the way that pagans exposed their babies and then frequented brothels where unknown incest must be frequent. But that they found both ways plausible shows that they didn't know which was true.