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When the Sanhedrin heard testimony from two independent, reliable eyewitnesses that the new moon occurred on a certain date, they would declare the rosh chodesh (first of the month) and send out messengers to tell people when the month began.
The problem with strictly lunar calendars is that there are approximately 12.4 lunar months in every solar year, so a 12-month lunar calendar is about 11 days shorter than a solar year and a 13-month lunar is about 19 longer than a solar year.
However, this does not necessarily mean that the universe has existed for only 5700 years as we understand years. Similarly, the Jewish calendar has different starting points for different purposes.
Many Orthodox Jews will readily acknowledge that the first six "days" of creation are not necessarily 24-hour days (indeed, a 24-hour day would be meaningless until the creation of the sun on the fourth "day"). The names of the months of the Jewish calendar were adopted during the time of Ezra, after the return from the Babylonian exile.
For a fascinating (albeit somewhat defensive) article by a nuclear physicist showing how Einstein's Theory of Relativity sheds light on the correspondence between the Torah's age of the universe and the age ascertained by science, see The Age of the Universe. The names are actually Babylonian month names, brought back to Israel by the returning exiles.
This process is sometimes referred to as "fixing" Rosh Hashanah. However, the Jewish New Year is in Tishri, the seventh month, and that is when the year number is increased.
If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look. This concept of different starting points for a year is not as strange as it might seem at first glance.