A door is not just a moving structure that can either block or allow access to another place. For many cultures, the door represents a portal or a gateway into new worlds, whether that it is in the literal or metaphorical sense. When we walk through a door and kiss the mezuzah, we are supposed to keep something in mind. What is that something, and even more interestingly, what does that have to do with the Exodus?
In addition to the mezuzah, there is something else about which we are reminded on a daily basis, and that is the Exodus of the Jews leaving Egypt. Although the Exodus is mentioned in more than one place in daily prayers, it most notably makes an appearance in the Shema. The reason why it's noteworthy is because in the Shema, the mezuzah is also mentioned. If I had to identify the single most important prayer for a Jew, it would be the Shema. It is a declaration of a relationship in G-d that translates into action. It is the prayer that a Jew recites twice a day, and is ideally the last thing a Jew recites while still alive. Shema encapsulates what it means to be a Jew.
If you want more of an idea of how the mezuzah and the Exodus are connected through Shema than merely being mentioned in the same prayer, look at the structure of the Shema, specifically the end of each paragraph. The first and second paragraphs both end with the importance of the mezuzah. The third paragraph of the Shema ends with that G-d brought the Jewish people out of Egypt. So what's the connection between the mezuzah and the Exodus?
Let's start at the verse that the Shema cites at the end of the third paragraph:
I am the L-rd, your G-d, who brought you out of the land of Egypt to be your G-d. I am the L-rd, your G-d. -Numbers 15:41
G-d didn't just bring the Jewish people out of slavery. If He did, we would only have to sing the first song of Dayenu and be done with it. G-d wanted to do so much more than that. As the verse in Numbers shows, G-d wanted to develop a relationship with the Jewish people. G-d wanted a relationship in which the Jewish people love G-d with all their heart, soul, and might (Deuteronomy 6:6). The way that the Jewish people develop that relationship with G-d is through Torah and doing mitzvahs.
That relationship did not really begin until the Exodus. Why the Exodus, and not something like the giving of the Torah at Mount Sinai? Why is the Exodus so special? It was the moment when the Jewish people went from being slaves to being a free people. This transition cannot be emphasized enough because serving G-d implies that there is free will. Without free will, having a Torah that instructs Jews to follow mitzvahs would be non-sensical. A lack of free will would also make moral judgments and ethics just as non-sensical. Free will is both at the essence of human dignity and of Judaism.
Prior to G-d bringing the Jewish people out of Egypt, the Jewish people were slaves. They worked all day, slept, woke up and did the same thing all over again. What they did as slaves was instinctive, routine, and void of any growth or progression. Upon walking through the metaphorical door, the Jewish people made that transition from slavery to freedom. This transition is so strong that Jews are to remember and re-live it every year at the Passover seder.
However, this is not something that we are supposed to relive only during Passover. Maimonides teaches (Mishneh Torah, Hilchot De'ot, Mezuzot 6:13) that the purpose of the mezuzah is to remind us of G-d's Oneness. This reminder of His Oneness is to arouse a love towards Him. Much like the Exodus, the mezuzah is a vital way in which the Jew develops a love of G-d that is just as transformative as what the Jews of yore must have felt when they saw the Sea of Reeds part. That sense of awe and gravitas is not something that is solely reserved for Passover. It is something that we can apply daily. By bringing the proper awareness and kavannah to the mitzvah of the mezuzah, we can relive the essence of the Exodus by not only literally walking through a doorway, but using the mezuzah as a portal to experience a relationship with the Divine.