Yoma, Chapter 3, Mishnah 2. In the previous Mishnah we learned how they would station a person to watch and see that morning had indeed arrived, in order to determine whether the time for slaughtering had come. According to one opinion, he had to say explicitly: "Harei p'nei chulam mizrach" - the entire eastern sky is lit up, even as far as Chevron.
The Mishnah's question: why is a watchman needed at all?
Our Mishnah opens by asking why this was necessary, and the commentators differ in explaining the question:
Rashi: why is there a need for a watchman to look out and confirm that it had become sufficiently light.
The Meiri: why is there a need for so much light, to the point that the entire eastern sky is illuminated as far as Chevron.
Either way, the answer is the same. The Mishnah relates that on one occasion the light of the moon rose, and from a general glance - without a watchman looking out toward the horizon - they imagined that the east had become light. On the basis of the moonlight, which they assumed came from the sun, they slaughtered the daily Tamid offering, but it was still night, and a Tamid slaughtered at night is invalid. Because of this, they had to take the animal that had been slaughtered as the Tamid out to the place where invalid offerings are burned, and they had no wish to see such a thing happen.
In practice, this incident did not occur on Yom Kippur, since astronomically the moon does not rise toward morning on the tenth of the month, when Yom Kippur falls, but rather it happened at another time of year. Still, since this was the mishap that occurred in that case, they were concerned that a similar problem and a comparable mistake could arise, and therefore they wanted a watchman who would tell them that dawn had indeed broken.
Bringing the Kohen Gadol down to immerse:
The Mishnah says that once the time was established, they would bring the Kohen Gadol down to immerse in a mikvah. Although the purpose of the immersion here, as we will see, is preparation for the service, the Mishnah shares with us an additional halachah regarding immersion in the Beis HaMikdash, and this was the rule in the Mikdash:
"Kol ha'meisech es raglav" - "covering one's legs" is an expression for relieving oneself, and anyone who did so was required to immerse in a mikvah.
"Kol ha'meitil mayim" - one who urinated was not required to immerse in a mikvah, but before he could continue with any activity in the Beis HaMikdash, he had to wash his hands and feet.
In summary: in this Mishnah we learned why a watchman was needed to testify that dawn had broken - out of concern for a mistake like the one that occurred when they took moonlight for sunlight and slaughtered the Tamid at night; and we also learned about bringing the Kohen Gadol down to immerse, and along with it the rule that applied in the Mikdash: one who relieves himself requires immersion, while one who urinates requires only washing of the hands and feet.