As we see from Yom Kippur’s morning Torah reading, a major focus of the practical mitzvot of the day is the service performed, primarily by the Kohen Gadol, in the Beit Hamikdash. This is a major part of the process of atonement of the day, even though we are able to receive a serious degree of atonement "on our own," by means of our repentance, our fulfillment of the mitzvot of the day, and itzumo shel yom, the day’s essence.
Regarding the individual’s observance of the day, by Torah law he has two sets of commandments: fasting and refraining from melacha (forbidden work). Keeping to these restrictions is both the fulfillment of a positive mitzva and the avoidance of transgressing a negative mitzva (see Rambam, Shevitat Asor 1: 1, 4). Interestingly, the prohibition of eating is written in the language of a positive commandment (do "innuy," usually translated as self-affliction), and the negative commandment is derived creatively. In the past, we have discussed halachic ramifications of this presentation. This choice of presentation and that of the interrelationship between the two sets of mitzvot also may indicate something about the philosophical nature of the mitzva to fast.
The Sefer Hachinuch (313) says that on Yom Kippur we are supposed to be spiritually minded, whereas eating drags us in the direction of physicality and even sin. The Sefer Hachinuch says the same basic idea about not working, which applies also to many special days during the course of the year. One could change the focus of this approach slightly and say that we are to positively act like holy people, indeed like angels, who do not eat or drink, etc. (see Avot D’Rabbi Natan 37).
Another approach is to view the refraining from eating and drinking as a personal element of service of Hashem, which parallels, to some extent, that which occurs in the Beit Hamikdash. This approach finds poignant expression in the Tefilla Zaka of the Chayei Adam, where we request that the lessening of our blood and fat should be considered as if we brought a korban and placed blood and fats on the altar. According to this approach, while the mitzva is performed by a lack of action, its nature is actually positive in nature. In other words, our fasting is parallel to the positive mitzva to bring a korban. If so, the overlap of this mitzva with that of refraining from work is not as direct.
Rav S.R. Hirsch (Vayikra 16:31) brilliantly connects these two elements found together in the following famous pasuk: "It shall be for you a complete day of cessation (shabbat shabbaton) and you shall afflict yourselves." He points that while other shabbaton’s are for Hashem, this one relates to the person himself. Not only does the Jew give up for a day his life of dominion through creative activity over his surroundings. He also acts in a way that conveys giving up his life, i.e., he refrains from eating and drinking, which are needed to live. In this way, the pasuk connects the elements and builds: act in a manner of extreme humility before Hashem, without viewing yourselves as deserving of all the rights He normally bestows upon us. May we make the most of all the spiritual opportunities of the day.