One might think that the obligation to recount the Exodus is from the beginning of the month (of Nissan). Therefore the Torah says, "On that day..."
Rosh Chodesh Nissan ― the first day of Nissan ― is one of the most significant days in Jewish history. It was then, in Egypt, that God gave the first mitzvah to the Jewish people ― the mitzvah of declaring the new moon and sanctifying the new month. For 2,448 years, God did so Himself, but now He entrusted the honor and responsibility to the Jewish people.
Why is this mitzvah so significant? Because it shows that we have the ability (and responsibility) to sanctify our lives and the passage of time. With this power to change and determine physical reality, we thereby become partners in the ongoing process of creation.
The First Day of Nisan is Rosh HaShanah for the following five matters:
1. Kings of Israel - They count their reigns from the First of Nisan, such that even if a King began his reign at the end of Adar, once Nisan began, it would be considered as the Second Year of his reign
2. Pilgrim Festivals - The Festival which occurs in Nisan, namely Pesach, is considered the First of the Three Pilgrim Festivals: Pesach, Shavuot, Sukkot. The application is that if a person donated a sacrifice, he has violated the prohibition of "Do not Delay" only if the time interval has passed defined by the three Festivals in the above order, such that if the donation was made, say, after Pesach, "Do not Delay" has not been violated until Shavuot, Sukkot, Pesach, Shavuot, and again, Sukkot, have passed.