The מלבי״ם (among many others) points out that the phrase אלה הדברים אשר צוה ה׳ לעשת is wrong if the pasuk is talking about שבת: שבת is all about the things you can’t do. The רמב״ן explains what the pasuk means:
We learn the laws of שבת from the work that was done in the משכן. שבת is our imitatio Dei, our re-enacting ה׳'s “resting” from creation. Why should the משכן be the source for what is defined as “creation”? The usual answer is that the משכן is a microcosm of creation itself; our building the משכן is re-enacting creation. The מלבי״ם has a long Kabbalistic essay on the symbolism of every part of the משכן in terms of the structure of the universe. But the connection is more textual than that. The phrases that the Torah uses in בראשית are used here as well:
Rabbi Leibtag points out that the command to build the משכן is given in seven דברות, culminating in שבת, exactly paralleling creation itself:
All of parshiot תרומה and תצוה are one long דברה that discusses pretty much all of the construction of the משכן and its components. The first day of creation was the day of actual creation ex nihilo; the rest of the week was forming what already existed in potential:
The second day was the creation of division with the separation of the מים מתחת from the מים מעל, corresponding to the דברה of the מחצית השקל. There is a custom of not getting married on Monday, since that is the day of the creation of מחלקת, but the מחצית השקל teaches us that not all חילוק is bad: the splitting of the שקל shows that no one is complete and we need different people with different skills and different perspectives in society. This is the מחלקת לשם שמים, what in modern coporate-speak is called “diversity”.
The connection between יקוו המים and the כיור is obvious and gives strength to our parallel.
The שמן המשחה was used to designate the holiness of the משכן and those who worked in it. We usually think of the מארות as the source of light in the world, but the Torah doesn’t see their primary prupose that way. They are לאתת ולמועדים ולימים ושנים. Just as the שמן המשחה serves to designate holiness in place and in people, the heavenly bodies serve to designate holiness in time.
Mickey Ariel suggested that what was really created on the fifth day was an ecosystem; with the creation of animal life that feeds or otherwise depends on one another . This corresponds to the קטרת , a mixture of spices combines to a harmonious whole:
The sixth דברה concerns the role of human beings in the construction of the משכן, corresponding to the creation of humanity itself.
And שבת is the culmination of creation, both of the universe and the משכן. But if we look carefully, not all the prohibitions of שבת can be easily derived from the משכן. The 39 מלאכות are a הלכה ממשה מסיני, not explicitly listed in the Torah, but they are hinted to in our parasha:
So the gemara sees three categories of מלאכה: the 36 “standard” ones from אלה, two from דברים, which I would propose are הבערה and כבוי, explicitly mentioned in our parasha (לא תבערו אש), and one unique מלאכה that doesn’t really fit in the idea of resting from creation: הוצאה. That needs its own derivation in the gemara:
Hirsch points out that הוצאה is fundamentally different. There are two reasons given for our observance of שבת: in the first record of the 10 commandments, it is (שמות כ:י) כי ששת ימים עשה ה׳ את השמים ואת הארץ. When Moshe recounts the 10 commandments, the reason is (דברים ה:יד) כי עבד היית בארץ מצרים ויצאך ה׳ אלקיך משם. As Hirsch sees it, רשות היחיד and רשות הרבים are societal constructs; they only exist as part of the rules that govern our relation to each other. What was created at יצאת מצרים was a new nation and a new society, עם ישראל, and it is specifically by refraining from הוצאה that we acknowledge ה׳'s role in that creation.
That is why Jeremiah emphasizes הוצאה as the aspect of שבת that preserves the Jewish people as a people:
So our parasha gives us a dual perspective on the observance of שבת: we acknowledge ה׳'s creation of the physical world around us, and in creating the nation of Israel with its unique role in that world. Perhaps that is why the gemara says (שבת קיח,ב) אלמלי משמרין ישראל שתי שבתות כהלכתן מיד נגאלים.