On the Shabbat before the Jews went out of Egypt, they took a lamb to be used for the Korban Pesach. The lambs were tied to the Jews’ bedposts and kept there until the 14th day of Nisan, when they were to be offered as Korbanot Pesach. Tosafot (Shabbat 87b) quote a Midrash Rabba, which states that when the Jews took their lambs, the first born of the nations of the world asked them what they were doing with them. The Jews answered that the lambs were the Pesach offerings for Hashem, who was about to kill the first born of the Egyptians. The Egyptians went to Paroh and their other leaders, asking them to let the Jews leave Egypt in order that the first born be spared from death. However, the leaders (with their hardened hearts) did not agree to let the Jewish people go free. The first born thereupon started a civil war against the other Egyptians and killed many of them.
The Maharsha quotes the Tur, who says that the teeth of the first born Egyptians were dulled and they were unable speak up. The Maharsha further explains that first born became concerned because the astrological sign of the lamb is the head and first of all the astrological signs. It was therefore also the sign representing the first born, who worshipped it as a deity. When they heard that the Jews intended to slaughter sheep, and, even worse, that their own deaths were predicted, they ran to Paroh demanding the release of the Jews, leading to the aforementioned civil war. The Ritva explains that on Shabbat HaGadol, when the Egyptian first born saw the Jews busy with their Korbanot Pesach and the prediction of their impending doom, they hurried to send the Jews out. The first born Egyptians were nervous because they knew no one would save them, so they killed many other Egyptians hoping the Jews would go free in the chaos and their own lives thereby spared.
This civil war is also mentioned in Tehillim 136, also known as “Hallel Hagadol”, which we recite every Shabbat and Yom Tov in Pesukei DeZimra. The Pasuk there says, “To smite the Egyptians with their first born.” If this Pasuk were referring only to Makkat Bechorot ( the tenth plague), it should have said, “To smite the first born of the Egyptians.” We see from this Pasuk that the first born waged war against the other Egyptians.
The miracle started on the Shabbat before Pesach, but lasted until the 14th when Bnei Yisrael brought the Korban Pesach. The Aruch HaShulchan comments that because it began on Shabbat, the Egyptians questioned why the Jewish people were doing labors that are usually forbidden on Shabbat (slaughtering). Perhaps this is why we have the custom of telling the story of Yetziat Mitzraim in the form of question and answer. Additionally, we know that Shabbat is dependent upon Pesach and vice versa; Hashem gave us the Mitzvah of Shabbat as soon as we left Egypt, even before we received the Torah. We say in Kiddush on Shabbat, “Zecher Liytziat Mitzraim,” recalling the Exodus from Egypt.
The Book of our Heritage (Sefer HaTodaah) adds that Bnei Yisrael needed a miracle because they took the lambs on Shabbat. They needed to give an explanation of this behavior to the Egyptians, but they could not lie (or even bend the truth to prevent possible arguments with the Egyptians) on Shabbat. Our Sages said, “Even an ignorant man would not tell lies on Shabbat.” They therefore needed a miracle to save them.