The relationship between the construction of the mishkan and Shabbos is two-fold. Notwithstanding its significance, work on the mishkan must cease when Shabbos arrives. Additionally, construction of the mishkan is the model for all prohibited categories of work. The foundation of the laws of Shabbos is that whatever was necessary to build the mishkan is labeled as a melacha on Shabbos.
There is no need for the Torah to clearly delineate what cannot be done on Shabbos. Each of the thirty nine melachos was in fact performed in the construction of the mishkan and each one is included in the all-encompassing phrase, "lo sa-a-se kol melacha - do not do any labor".
Chazal single out the prohibition of carrying as being unique. A special source is necessary to teach us that carrying is actually prohibited on Shabbos and the rishonim question why this is so. If carrying was done as part of the process of the mishkan's construction it should obviously be prohibited, no less than the other thirty eight melachos. Tosafos (Shabbos 96b) explain that carrying is a "mleacha ga-ru-ah - inferior melacha" and thereby needs an independent source to prohibit it. What is unique about carrying that makes it "inferior"? The Ohr Zaruah explains that unlike the other melachos, carrying doesn't change the item; the object is merely transported from one place to another. If carrying is so radically different from the other thirty eight melachos, why does the Torah prohibit it? Is there a unique dimension of Shabbos that can be derived from this unique melacha?
Shabbos is referred to as both a "zecher l'ma'aseh Breishis" and a "zecher l'yetzias Mitzrayim"; it simultaneously commemorates the creation of the world and our leaving Mitzrayim. These two events highlight two aspects of the relationship Hashem has with the world, namely Hashem is both the Creator and the One who controls the world. Thirty eight melachos highlight His role as Creator. We refrain from creative activities thereby acknowledging Hashem as the "Boreah shomayim va'aretz - Creator of heaven and earth". Not carrying is not a confirmation of creation, but rather a declaration that Hashem is the One who controls the heaven and the earth. When Yosef is appointed by Pharoh to control the land of Mitzrayim, he is assured that without permission granted by Yosef no one will be permitted to lift their hands or feet- "lo yarim ish es yado v'es raglo bechol Eretz Mitzrayim" (Breishis 41.) Total submission to a ruler includes the acknowledgment that permission must be granted to "move" anything, because only the ruler is in control. On Shabbos we commemorate yetzias Mitzrayim by refraining from carrying unless permitted to do so according to the intricate laws of Eruvin. Whereas thirty eight melachos are a zecher l'ma'aseh Breishis, observing the thirty ninth melacha of carrying declares our commitment to zecher l'yetzias Mitzrayim.
Both dimensions of Shabbos are fundamental to our entire service of Hashem. One would have expected that priority should be given to acknowledgment of Hashem as Creator which preceded His role as the One who took us out of Mitzrayim. However, the aseres hadibros begin with Hashem as the One who took us out of Mitzrayim. The mefarshim explain that only through yetzias Mitzrayim did we come to recognize Hashem also as our Creator. Nobody witnessed creation but our actual experiencing yetzias Mitzrayim enabled us to accept Hashem as the One who controls the world. The corollary to yetzias Mitzrayim is creation. Only the Creator of the world can control it as the events of yetzias Mitzrayim indicated. In the realm of the laws of Shabbos the melacha of carrying teaches us this same lesson. Perhaps this is why the introduction to Shabbos was given to the Jewish people in the context of carrying. The first melacha was not to carry the mon in the desert. The Jewish people were introduced to Shabbos via their experience at yetzias Mitzrayim. It was only through the vehicle of zecher l'yetzias Mitzrayim did they eventually realize Shabbos as the zecher l'ma'aseh Breishis. Even Maseches Shabbos which encompasses all of the laws of Shabbos begins with the topic of carrying. It is the commitment to the proper observance of this melacha that enables us to eventually observe all of the halachos of Shabbos. We thereby affirm our complete belief in ma'aseh Breishis and yetzias Mitzrayim, the two fundamentals of Shabbos observance and the foundation of our entire service of Hashem.