A quick check of WP says that the largest meta-study found no correlation, they found what they claimed to be "statistical errors" in the 20 studies that did find a correlation. To me that doesn't say "no", it says the evidence we have to date is weak (unless there's a much larger set of "error free" papers backing the "no correlation" side).
Personally I think there is a grain of truth in it, and my ex-wide who worked in ER for several years (as a cleaner) swears by it. However the moon doesn't actually create freaks from otherwise normal people, rather it provides just enough light for existing freaks to wander about at night and hurt themselves/others. I wouldn't expect the effect to be noticeable in a modern city hospital because city lights outshine the moon anyway. Weather also plays a huge role in the number of freaks wandering around at night, like everyone else, they (mostly) have enough sense to get out of the rain and are much more active on warm nights than cold. According to my ex-wife and her co-workers, a full moon on a hot and humid Saturday night is the perfect freak storm in a rural (or beachside) ER, statistically you're more likely to be murdered on a hot humid day than any other day.
There's a very good reason that virtually every ancient religion has a sun and/or a moon god, they were observed to rule the natural cycles around them. Modern city/urban life obscures most of those observations and people are left wondering why the hell stone age people went to the effort of building places like stone henge.
I think it's pretty much the same reason our modern society went to the effort of building the LHC or the Hubble telescope, ultimately they were trying to understand the world around them. Knowing where and when to turn up for an "all you can eat" buffet is a very deep behaviour in evolutionary terms, creatures as diverse as apes, jellyfish, corals, bears, and crocodiles make good use of it.
In most cases we are at a complete loss when it comes to explaining these things in detail. For example, how do crocodiles "know" to gather an hour or so before fish become trapped on a flooded river ford? - The brief event (filmed by Attenborough) only happens right at the peak of a king tide. AFAIK, nobody (including Attenborough) has a clue how the hell the crocs tell the difference between a high tide and a king tide BEFORE it arrives. In ancient times people just accepted that (say) the crocs inexplicable ability to predict king tides was due to "divine knowledge".
Humans are the undisputed masters of observing and exploiting patterns, however the root cause of the pattern is often irrelevant to it's utility. Ancient people would have simply observed the crocs (a wise defence behaviour anyway) and been alerted to the imminent fish bounty by their behaviour, some would have been mauled/taken by crocs when they went after the fish, lucky escape stories would abound, semi-random rituals would rapidly emerge to appease and thank the crocs. Next thing you know everybody wants a row of granite crocodile gods adorning their pyramid's driveway, and virgins are feeling nervous.