What does "Asher Hikriv Lo Yih'yeh" come to preclude?
Rashi: It comes to preclude a T'vul-Yom, a Mechusar Kipurim and an Onan (they cannot offer) from receiving a portion in the skins [of Kodshei Kodshim]. 1
See Sifsei Chachamim.
Why does the Torah teach here that the Kohen receives the skin of an Olah?
Ramban #1, Moshav Zekenim #1: The same applies to a Chatas and Asham; it is written here, before Shelamim, because the owner receives the skin of a Shelamim.
Ramban #2, Moshav Zekenim #2: The Kohen receives the meat of Chatas and Asham; surely the skin has the law of the meat. The Torah needed to teach about Olah, for he does not get its meat.