A new month does not automatically begin when the moon reappears. Rather, the judges of the Beit Din sanctify the month, as it says, “This month shall be for you” (Shemot, 12:2). HaShem showed Moshe Rabbeinu the configuration of the moon in its renewed state and said, “This testimony shall be given over to you” (Rosh HaShanah 22a). That is, witnesses must come before you (the judges) and testify that they saw the new moon, and you shall sanctify the month based on their testimony.
After Moshe Rabbeinu’s death, the authority to establish the Jewish calendar was conferred upon the high court of every generation, on condition that its judges had received rabbinic ordination (semichah) in an unbroken chain from Moshe Rabbeinu; and with the stipulation that such ordination can only be transmitted in Eretz Yisrael(Rambam, Hilchot Sanhedrin 4). If a time comes when the Jews are unable to fix the months by way of a Beit Din, the halachah states that they must do so using mathematical calculations.
Thus, even though the lunar cycle is a natural phenomenon, the renewal of the moon does not, by itself, sanctify the month. Rather, the Jewish people consecrate the months, and they cause the holiness within time to be revealed. This explains why our Sages decided to end the central blessing of the Musaf prayer on Rosh Chodesh with the words “Blessed are You, HaShem, Who sanctifies Israel and the beginnings of the months” (Berachot 49a). Perhaps this is also why the first mitzvah the Torah commanded the Jewish people was the mitzvah of sanctifying the new moon (Shemot, 12:2), for this mitzvah displays a facet of Israel’s unique sanctity – our ability to reveal the holiness within time.
 If witnesses who saw the new moon on the night of the thirtieth come before the Beit DinBeit Din the next day, the court would sanctify the month on that day, transforming the thirtieth day of the previous month into the first day of the new month, Rosh Chodesh. They would then immediately offer up the special Rosh Chodesh sacrifices. Consequently, the previous month would become an incomplete one, having only twenty-nine days. If, however, no witnesses arrive on that day (the thirtieth), it is clear that Rosh Chodesh will be the next day – the thirty-first – and the previous month was a full one, consisting of thirty days. In such a case, there is no need for the Beit DinBeit Din to accept witnesses or declare the beginning of the new month, for in any case there are only two possible days on which Rosh Chodesh can fall, and if no witnesses come on the first day, Rosh Chodesh will [automatically] fall out on the next day (Rambam, Hilchot Kiddush HaChodesh 2:8).