What does the opening phrase ("ve'Achlu osam ... ") in the Pasuk mean?
It is unclear what 'mundane activities' refers to - unless it pertains to the future.
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 the middle phrase ("Lemalei es Yadam ... ") mean?
Rashi: It means that it is via the Korbanos of the Milu'im that Aharon and his sons would be inaugurated for the Kehunah and become sanctified to serve as Kohanim.
Refer also to Shemos 28:41: 1, 2 & 3.
And what can we extrapolate from the final phrase ("ve'Zar Lo Yochal ki Kodesh heim"?
Rashi: It is a warning forbidding a Zar against eating all Korbanos that are Kodesh Kodshim.