Carrying Between Private and Public Domains
The Torah forbids carrying between a public domain and a private domain.
On the one hand, this seems like the most insignificant of all the thirty-nine forbidden labors; it is hard to see exactly what constructive outcome is achieved. The Tosafot (beginning of Shabbat) call it an "inferior labor."
Yet in our sources it seems to be by far the most important labor of all: in tractate Shabbat itself it occupies far more space than any other labor, and on top of that there is an additional tractate, Eiruvin, which is devoted almost exclusively to this labor. Likewise the Shulchan Arukh has an entire section devoted toeiruv, which is related to making carrying permissible.
Indeed, tractate Shabbat begins with this melakhah, through the example of a rich man within the house and a poor person standing without. Each one is capable of either bringing in or taking out objects in a forbidden way.
Rav Natan of Breslav explains that the essence of this melakha is BERUR, or clarification. The private domain, in Hebrew "the domain of the one", represents the domain of the One, the realm of holiness and God's oneness. The public domain, "domain of the many", represents the realm of division and discord, like idolatry, which acknowledges a variety of discordant forces in the world. Carrying from one to the other represents a kind of clarification of values. This type of clarification is the very essence of our mission as Jews: to know to find our way in the confusion of the material world and know to distinguish between those aspects which can be elevated to holiness and those which are inherently far from holiness and need to be distanced.
Rav Natan further explains that the aspects of taking out and bringing in represent two ways of serving Hashem. A person who is "inside", in the domain of holiness, is rich in spirituality, but he is sometimes obligated to venture out into the perilous "domain of the many" and face the trials of material existence, for example those of the workplace, in order to find the aspects of holiness found in mundane activities. Conversely, the person who is outside, enveloped in worldly concerns, is impoverished; he is obligated to find times to venture in, to find refuge in the study and observance of Torah and in the presence of righteous individuals.
But Shabbat is the day when we view the world as perfected, not only materially but also spiritually, so on this day we refrain from carrying. - (Likutei Halakhot Shabbat 7:30)