Doe anyone know the reasoning for the requirement to use a flame that was lit over Yom Kippur for Havdala?
Just like on Shabbos, after Yom Kippur we make a Bracha on fire since we were not permitted to use fire before, and it is like a new entity to us.
After Shabbos we may use new fire because this is how fire was initially brought into the world after Shabbos Adam took two stones and made fire and made the bracha.
But this does not apply after Yom Kippur, so we must make a bracha only on fire the rested throughout the day of Yom Kippur.
4. However, a blessing must be recited over fire even when [Yom Kippur] falls on a weekday, [thereby indicating that] it is a new entity from which we did not benefit on Yom Kippur, for it is forbidden to light fire [on that day]. Therefore [the night after Yom Kippur] is different from the night after any other festival, when a blessing is not recited over [fire].
[The rationale is that] a blessing is recited over fire only at the time when it was brought into being, i.e., on the night after Shabbos, when fire was first created, or on the night after Yom Kippur, when fire is like a new entity for us, as was explained above.
5. On the night after Yom Kippur, the blessing is not recited over fire that was produced [that night] from flint or wood, nor over fire that was kindled from such fire, but only over fire that "rested" from work, such as a lamp that has been alight since the eve of Yom Kippur.
On the night after [any] Shabbos, a blessing may be recited over fire that was produced from flint or wood, because it was in this manner that fire was originally created on the night after [the first] Shabbos. As [our Sages relate, on the night after the first Shabbos of creation] Adam took two stones, struck them together, brought forth fire, and recited the blessing borei meorei ha'esh, [praising G-d as "the Creator of the lights of fire"]. Hence on the night following [any] Shabbos, the blessing may be recited over any fire (unless it was used for forbidden labor on Shabbos, as stated in sec. 298[:7]) - even over fire produced at that time from flint, for this resembles the way it was originally brought into being.
On the night after Yom Kippur, by contrast, the reason for this blessing is that one was unable to benefit [from fire] on that day, and now this is permitted. Hence, the blessing should be recited only over fire that actually "rested," i.e., that was kindled on the day before Yom Kippur and "rested" throughout the day of Yom Kippur. Even when Yom Kippur falls on Shabbos, the blessing should be recited - at night, at the close of the holy day - only over fire that was kindled on the day before Yom Kippur and "rested" throughout the day of Yom Kippur, to show that this day was sanctified in that fire was not kindled during that time.
TorahLab give the following explanation:
On Yom Kippur we are celebrating the fact that we hadn’t used fire all day. In this way Yom Kippur is different from all other Jewish holidays, on which the use of fire is permitted. So it is therefore appropriate to use a flame that had ‘rested’ all day. It had been burning before Yom Kippur, had not been used over the holiday, and had now become permitted once again.
There's more explanation there too.