Bearing in mind that the redemption from Egypt took place on the fifteenth of Nisan, when Yisrael left Egypt, why does the Torah write here that Hashem saved us, "va'Yosha' Hashem ba'Yom ha'Hu," meaning on the twenty-first?
Seforno: Because as long as the Egyptians were chasing them, their status was that of fleeing slaves, and it was only now that they (the Egyptians) were dead that they were truly redeemed.
What exactly, did Yisrael see "Al Sefas ha'Yam"?
Rashi: They saw the corpses that the sea spat out onto its banks. 1
Rashbam and Hadar Zekenim: Yisrael, who were already on dry land, actually saw the Egyptians drowning in the sea. 2
Targum Yonasan: They saw the Egyptians dead yet not dead, cast on the sea shore (half-dead, and Yisrael witnessed their death).
Rashi: Because their Emunah was weak and they thought that, just as they had emerged alive from the sea on one bank, so too, did the Egyptians emerge alive from the sea on the other bank, and were about to chase after them. So the sea spat out the corpses as proof that the Egyptians had been killed.( It is difficult to comprehend how, even as they witnessed one of the greatest miracles to have ever taken place, and despite Moshe's assurance of Hashem's salvation (See Pesukim 14:13 -14), Yisrael had doubts.)
Rashbam: It is as if the Torah had written "va'Yar Yisrael (Asher) Al Sefas ha'Yam, Es Mitzrayim Meis."
"Va'Yosha' Hashem" - Why is this salvation explicitly associated with Hashem's Name (Havayah), more so than other salvations?
Maharal (Gevuros Hashem, beg. Ch. 47, p. 179): Hashem had decreed four hundred years 1 of exile upon the descendants of Avraham. By human understanding, and even from the perspective of the Mal'achim, all the years when Avraham was alive and living in Eretz Kena'an, should not be counted towards that number. They experienced no hardship at that time! And yet, Hashem decided to count the 400 years starting from the birth of Yitzchak - during Avraham's lifetime. In Hashem's view, this indeed could be counted as "strangers in a land not theirs," even if they felt it at the time in just some small aspect. Hashem's justice is very exacting, in accordance with His great wisdom. 2
The Midrash (in the Yalkut) tells that in Heaven, Uza (Sar of Mitzrayim) argued to Micha'el (Sar of Yisrael), that Mitzrayim still had rights to enslave the Bnei Yisrael through the end of the term of 400 years. Micha'el was silenced - until Hashem Himself answered Uza. Thus, it seems that Chazal are interpreting the word "Va'Yosha'" in the legal sense; that Hashem argued in their defense. Micha'el had nothing to say; and only Hashem was able to argue on their behalf. Also see Otzar Midrashim - Why does our verse say "Hashem saved the Bnei Yisrael, on that day, from the hand of Egypt"? We had already left Egypt, and we were no longer under their control! Rather, we were about to lose to Egypt in the Heavenly Court, until Hashem argued on our behalf.
As Chazal put it, "Hashem is exacting with the righteous, as much as a hair's breadth" (Yevamos 121b). Maharal adds - Just as the decree itself (of 400 years' exile) was brought about by exacting justice - for Avraham having said, "By what shall I know that I will inherit [the Land]?" (Bereishis 15:8); so too was Hashem lenient to calculate the fulfillment of that punishment by such standards, by counting the Avos' years in the Land towards the years of exile.