What was the point of reminding them of what they had seen?
Rashi: Hashem was pointing out to them that His connection with them was not indirect, 1 because it was with their own eyes that they had witnessed what Hashem had done to the Egyptians, how their sins had accumulated already before they started up with Yisrael, yet Hashem had only punished them on account of what they did to them (Yisrael).
Seforno: He was highlighting His Midah of compassion - since Yisrael had seen firsthand how He pleaded with the Egyptians over and over again to repent, before punishing them. 2
What is the meaning of "I carried you"?
Rashi #1: This refers to the morning of the fifteenth of Nisan, when Hashem gathered Yisrael, who were scattered all over the land of Goshen, to Ra'amses before leaving Egypt.
Rashi #2 (12:37): Hashem transported Yisrael from Ra'amses to Succos, a distance of 120 Mil, in no time at all. 1
Targum Yonasan: It refers to the night of the fifteenth of Nisan in Egypt (when Yisrael were forbidden to leave their houses, and) when Hashem transported them first to Ra'amses and from there to the location of the Beis-ha'Mikdash to do the Korban Pesach, and returned them. 2
Why does it say "on the wings of eagles"?
Rashi: Because, unlike other birds, who, afraid of the eagle who soars above them, hold their young between their legs - thereby leaving them at the mercy of archers who shoot at them from the ground, eagles carry their young safely on their backs - so that, should the archers shoot at them, the arrows will piece them and not their children. Much in the same way, on the night of Keri'as Yam-Suf, Egyptians fired arrows and slingshots at them, and the Pillar of Cloud containing the Shechinah (Kevayachol), moved in between the two camps and absorbed the projectiles, lest they hit Yisrael.
Seforno: Just as eagles fly high in the air along a route that no other birds venture, so too, did Hashem take Yisrael out into the desert, along a route that no other man would venture, in order to separate them from the abominations of Egypt and their practices, to become His nation.
What did He mean when He said "v'Avi Eschem Elai"?
Rashi: I brought you to worship Me." 1
Ramban and Targum Yonasan: I brought you (from Ra'amses - Targum Yonasan) to Har Sinai 2 to receive the Torah.
Seforno: I brought you to the mountain of Hashem which has been designated for prophecy.
Moshav Zekenim: It is an expression of marital relations. Hashem was Mekadesh Yisrael through all three ways
To be for you a G-d (Rashbam).
Where the Shechinah was currently located (Ramban).
Presumably, this refers to the Luchos, which are called Edus, and were written with His finger (below, 31:8); this is like a signed document (refer to Bereishis 48:9:151:1). The Torah was not written yet, and it was not in Hashem's handwriting. Elsewhere, the Luchos are compared to the Kesuvah (refer to 34:1:3:1*, 19:1:151:1**)! (PF)
Gur Aryeh (17, DH uv'Midrash): A Midrash says that Hashem forced Yisrael to accept Torah, so the law of a rapist would apply, and He can never divorce them. Even though He did not sin, since the Chibur was totally due to Him, without the recipient's Da'as, it is permanent.