Adam, the First Man, and Chava, the First Woman, were created on the last of the Six Days of Creation, the Eve of Shabbat - This day is celebrated as Rosh HaShanah.
Thus, when we say on Rosh HaShanah "Today is the Birthday of the World," we mean that it was the Date of Creation of the first Human Beings.
The Midrash (VaYikra Rabbah 29:1) tells us that on the very first Rosh HaShanah Adam and Chavah committed the very first sin. G-d judged them and forgave them. G-d said, "Just as you were judged before me on this day and emerged forgiven, so will your children be judged this day and emerge forgiven."
For this reason, G-d sees fit to judge all the people and nations of the world every year on this day.