1) Because it is the first word on the pasuk and deserved a zakef. At this distance from the etnachta, this would be a segolta. But a segolta needs a preceding zarka, and this is the first word. And so it becomes a shalshelet.
2) Rav Chaim Kanievsky, in Taama deKra, cites a different sefer, which gives a consistent explanation of shalshelet as extension. And he says:
וישחט [in Vayikra 8:23], because he extended in the slaughtering of the peace-offering more than for the burnt-offering or the sin-offering, in order that much blood would come out, for he [=Moshe] needed to sprinkle from it on the ear-tips, thumbs, and toes [of Aharon and his sons].
3) As mentioned in Menachem's answer, Ibn Caspi writes that that it connotes indecision. In terms of msh210's comment wondering what Ibn Caspi says about Moshe, alas, Ibn Caspi writes extremely tersely in the beginning of Vayikra, which is where the relevant pasuk occurs, so we don't know precisely what he would say.
4) Finally, there is this:
I found a beautiful elaboration of these four occurrences in "Jewish Thought - a Journal of Torah Scholarship," published by the OU in conjunction with Yeshivat Ohr Yerushalayim, and edited by Rabbi Moshe Chayim Sosevsky. The specific essay, "The Shahshelet Cantillation," was written by Mois A. Navon, and appeared in Volume 4, No. 1, 5755-6. Three "Shalshelaot" appear in Bereshit; one in Vayikra, in the Parshah we read this Shabbat. Navon discusses each.
Tanya Shemini:3 illustrates Moshe's behavior and attitude at that time:
"Moshe said (to Aharon), 'So has the Holy One, Blessed Be He said to me to appoint you Kohen Gadol.' Aharon said to him, 'You toiled in the Mishkan, and I am made Kohen Gadol?!' (Moshe) said to him, 'By your life! Even though you have been made Kohen Gadol, it is as if I have… For just as you rejoiced in my greatness, so I rejoice in your greatness.' "
The Shalshelet is placed upon the final act of "Shechitah," "Ritual Slaughter," performed by Moshe. Navon explains beautifully the full meaning of the Shalshelet. "The first rung of the Shalshelet suggests his anguish; the second, his resolve and forbearance; and the third, the greatness of his humility and self-sacrifice."
5) As a postscript, to address the edit in the question:
After I put in the reference to Wikipedia, I saw a reference to this explanation that suggests that Moshe was feeling internally conflicted because he wanted to be a Kohen himself. That explanation claims to come from Rashi, but I just checked again, and Rashi on Chumash there doesn't say a word. Anyone else have a reference backing this up?
I didn't find it in Rashi. However, see this blogpost by Rabbi Ari Kahn, who notes a midrash in Vayikra Rabba {perhaps cited by Rashi} that
R. Helbo said: All the seven days of consecration Moshe ministered in the office of Kohen Gadol, and he imagined it was his. On the seventh day He said to him: ‘It belongs not to you but to your brother Aharon.’ This is [indicated by] what is written, "And it came to pass on the eighth day, that Moshe called Aharon and his sons and the elders of Israel, and he said unto Aharon…" (Vayikra Rabbah 11:6)
and this is connected to Moshe's earlier improper statement of "Send, I pray Thee, by the hand of whomever You will send (Shmot 4:13)". Rabbi Khan suggests that Moshe would naturally have felt disappointment in this. And so, he connects it to the hesitation indicated by the shalsheles.
