Background

Today science assumes, that the moon is the common "child" of the earth and a planet of the size of the mars 4527 million years ago. This child, the moon, influences in many ways life on mother earth. Low tide and high tide are the most obvious impacts. Two perhaps less known phenomena: exactly as the daily postergation of the rise of the moon  by 50 minutes also low tide and high tide arrive 50 minutes later every day. During full and new moon there are spring tides, during half moon neap tide(transition from 1. to 2. quarter, respectively from the 3. to the 4. quarter) .

The moon also plays an important role in the formation of live on earth. Even today he makes sure, that the trees and plants we know can grow. For human beings a world without moon would be inhabitable, since the moon is a great climate regulator by influencing the rotation of the earth. Without the impact of the moon the earth would begin to totter in space. There would be no regular season, the climate would change abruptly and thus destroy our life basis. (like on the Mars)  Without moon the day on earth would only have eight instead of 24 hours.
Today manifold impacts on animals and plants can be observed, to the point of moonstruck humans. In most cultures the calendar was based on moon phases until the introduction of the sun calendar. Our word month gives evidence of that.

Moreover part of our bank holidays depend on the moon. Since the year of 325 after Christ, the council of Nicea, Eastern has always been celebrated the first Sunday after the first spring full moon. Most of the traditional feasts in Asia are also celebrated according  to the moon calendar. The Buddhist Vesakh feast, which is dedicated to the birth of Buddha is traditionally commemorated during the first May full moon. In parenthesis: The fact that in the year 2000 the birthday of the Pope and Buddha coincided on the full moon day hopefully will promote the inter-religious dialog - even when the moon calendar  "says" nothing about it.

Nature Rhythms

Many moon rules refer to rural live. Sowing and harvesting, logging but also nature - oriented curative treatments for example with medical herbs.
This is not amazing, since country lore and moon calendar - same as astrology with which exist some overlaps - derive from a time, when also people of our lines of latitude depended more directly on the rhythms of nature than in the engineered world of today. Most people depended directly or  indirectly existentially on agriculture, so that the exact observance of nature was a matter of life or death. They weather proverbs give evidence of the quality  of the observations. In many cases the they stand scientific verification and reach  a hit rate between 60 and 80 % partially over 90 % which is good in comparison with the hit rate of long term forecasters of today reaching 70 %.

It is difficult to decide, if the moon calendar describes directly the impacts of the moon's power or the moon is more a finder of the heavenly clock who visualizes the time and its impacts but does not make them. Finally it is irrelevant.

New popularity
In our lines of latitude the moon calendar became very popular in Europe again by the book &cit;Guided by the Moon - The Application of the Lunar Calendar in Everyday Life&cit; written by Johanner Paungger and Thomas Poppe.
They meanwhile have published a whole series of books about the influence of moon rhythms on different areas of daily life.
Many of these rules (and more) are taken into account in MoonSignCalendar.net.

Recommendations and admonitions in the moon calendar
For many activities and projects exist moon rules. Applying them, you can identify the days when the moon is favorable. From the perspective of the moon sign calendar all other days are less apt. For some projects also exist direct negative rules: Often it is a mixture of both, like: "Waning moon (positive) except on water days "
Difficult is the handling of admonitions for operations and other important projects.
In the relevant moon literature they say, that there are other rules for emergency operations etc. In these cases it appears more reasonable though to emphasis the especially recommendable days in place of the critical ones. Otherwise there exists the risk, that the admonition turns into a self-fulfilling prophecy and the negative impacts are stronger than the power of the moon. Anyway, it seems to be true, what the "gurus" of the moon calendar wisely admit: As to risks and side effects question you common sense.