The Ramban (Nachmanides) quotes a posuk in Shemos that states:
that thou mayst know that there is none like me in all the earth.
The Ramban explains:
This statement is a positive commandment, as it says, "I am Hashem, who teaches and commands you that you should know and believe that there is a Hashem, and He is your God," that is to say, He existed before, from Him came everything by means of his desire [i.e., He created the world because He wanted to, not by accident] and capability, and He is your God, so you are required to serve Him. It says in the text, "who took you out of Eretz Mitzrayim," for He took you out of there to teach you about the reality and the desire [of Hashem], for with His knowledge and providence we left there, and to teach about the newness, for from the beginning of the world, not a thing has changed from His nature, and to teach you about what He can do, which will teach you about His uniqueness, like it says above [in Shmot 9:14], "so that you will know that there is no one like Me in all the land." And this is the reason for [the phrase] "Who took you out" - for they know of and bear witness to all of these [ideas.]
The Or HaChaim HaKadosh writes that it is to teach us that we were in a state of impurity, and that G-d, the true one G-d, took us out of it:
אנכי ה׳ אלוקיך, "I am the Lord your G'd, etc." The reason G'd repeats "who took you out of Egypt, the house of bondage," was to heighten the contrast between then and now. The Israelites had been enslaved in a country which was notorious for not releasing slaves nor letting them escape (compare Isaiah 14,17). The Israelites had two major strikes against them at the time. They were mired deeply in impurity, and they were under the rule of a king notorious for not letting anyone escape. (emphasis mine)
The Or HaChaim further writes:
G'd manifested Himself in this dual capacity to remove all doubt about philosophies which inspired idol worship by suggesting that He had partners. G'd's taking the Israelites out of Egypt should have dispelled any such doubts about any opposing deities possessing any power at all. מבית עבדים, from the house of bondage. G'd makes the point that since it was He Who liberated us from one master, He is entitled to demand that we show Him obeisance. (emphasis mine)
So, based on the above, we can learn a few things:
- G-d is the One that has the supervision. G-d is the true one G-d, despite of what the theological mindset was from the Egyptians. They believed in deities. We believe in the true One G-d.
- G-d has the power to transform a nation to impart purity and holiness into it.
The Kedushas Levi explains why it is a fundamental principle. The Kedushas Levi explains that in order for G-d to give the Jewish people His Torah, they should have been released from the bondage of Mitzrayim and pharao:
The basis for this exegesis appears to be that G’d did not give the Torah to the Jewish people until after the Exodus from Egypt because there is a difference between keeping the Torah out of fear (of punishment) and keeping the Torah out of love for G’d. People who keep the Torah out of a feeling of love for G’d are called: בן, “son,” as in בני ישראל, “the Children of Israel,” whereas people keeping the Torah out of fear are still considered עבדים, “slaves.” Had the Israelites received the Torah prior to their redemption from Egypt they would have accepted it out of fear, so that G’d could not have taken them out of Egypt legally, as they “belonged” to the Egyptians, having been their slaves. Having received the Torah after the Exodus, when they were free men, they certainly had not been under duress in accepting it, but had done so out of love for the G’d Who had redeemed them. G’d therefore is explaining in the verse above that the time had come now when He was able to give them the Torah in their capacity of their being His children.
In this answer I cite different sources on what it means that everyday, we must see ourselves leaving Mitzrayim. Leaving Mitzrayim means the release of the divine soul from the "serpent's skin"- the bondage. And thus, being able to receive G-ds Torah and absorb it.