The Shulchan Aruch (OH 611:2) lists two surprising leniencies regarding the laws of Yom Kippur:
1) On the afternoon of YK, one may prepare vegetables and crack nuts in preparation for the meal after the fast. (Note: this was the original halacha, but the Shulchan Aruch also says our current custom to prohibit it.)
2) If a fire breaks out on YK, one may rescue enough food from it in order to suffice for one meal after the fast. (By comparison, on Shabbat one could rescue enough food for each of the remaining Shabbat meals.)
Both of these seem to contradict the principle that you cannot prepare on Yom Tov for after Yom Tov. So why are they permitted?
I think they can best be understood using an explanation borrowed from the laws of Shabbat.
The Rambam (Hilchot Shabbat 24:12) points out that the purpose of Shabbat would be lost if all that was prohibited was technical 'melacha'. People would still be able to go about most of their business like on every other day, and the idea of resting on Shabbat would be lost. Therefore the prophets and/or rabbis added a bunch of laws, for example muktza, and honoring and enjoying Shabbat, in order to make the day different and "restful" compared to the remainder of the week. They even added halachot which are not performed on Shabbat, but which contrast with Shabbat and thus make the day more special. Among these are havdalah; preparation for Shabbat; and the prohibition on eating a meal late Friday afternoon. By having to work before and after Shabbat, and by not eating before Shabbat, the resting and eating on Shabbat itself stand out in greater contrast.
The Torah's prohibition of work on YK seems to allude to a similar idea.
No 'melacha' shall be done on this day, for it is a day of atonement, to atone for you before Hashem your God. (Vayikra 23:28)
There is no obvious connection between work and atonement. So why is atonement mentioned here as the reason for not working? The connection is hard to understand.
I think the explanation is that atonement is what makes the day special and unusual. And like all other special days, we refrain from work on it, in order to distinguish the day from normal days. Atonement causes specialness, which in turn causes the prohibition on work. There is a connection between work and atonement, but it is through the intermediate concept of making the day distinguished and unique.
This concept of specialness seems to be exactly what we have seen regarding Shabbat. Just as we are commanded to work before and after Shabbat, it seems we are supposed to eat before and after YK. The requirement to eat before YK is well known. The permission to prepare (in certain ways) on YK for a post-fast meal indicates that this meal is integral to the day, even if it is not technically required.
Perhaps this suggests that the purpose of fasting on YK is not to make us unhappy (as might be the case with Tisha Beav). The importance of eating immediately before and after the fast contradicts the idea of maximizing discomfort. At the same time, it increases the sense of unusualness, as the contrast between how we live on YK and on "normal" days is heightened. It seems to me that we fast to create a sense of urgency, a feeling that we cannot go about our lives as we have until now. When the time comes to change ourselves, it is easiest to do so when the circumstances we are living in change drastically as well.