What does the opening phrase ("ve'Achlu Osam ... ") in the Pasuk mean?
Rashi: It means that Aharon and his sons eating the Eil ha'Milu'im will atone for all 'Zarus' (any Avodah performed by a Zar) and abominations of which they (the Kohanim) are guilty. 1
The Gemara in Pesachim, 29b, learns from here that when the Kohanim eat the meat of a Korban Chatas, the owner receives his atonement. In the current case, the owners happened to be synonymous with the Kohanim. Moreover, we learn from here that eating Kodshim is a Mitzvah (Rashi, Pesachim, 72b).
What does "Lemalei es Yadam ... " mean?
Rashi: It means that it is via the Korbanos of the Milu'im that Aharon and his sons would be initiated for the Kehunah and become sanctified to serve as Kohanim.
Refer also to Sh'mos 28:41: 1-3.
What can we extrapolate from the final phrase ("ve'Zar Lo Yochal ki Kodesh heim"?
Rashi (in Yevamos, 32b): This is not synonymous with the La'av of Me'ilah, since, whatever Kohanim are permitted to eat is not subject to Me'ilah.
Makos, 18b: The :La'av of Zarus only applies however, to a Zar who eats Kodshei Kodshim after the Zerikas ha'Dam, when Kohanim are permitted to eat them