The Gemara Gittin 59b states that there's a Biblical command to respect Kohanim by giving them 'first rights':
תנא דבי רבי ישמעאל וקדשתו לכל דבר שבקדושה לפתוח ראשון ולברך ראשון וליטול מנה יפה ראשון
Tanna Debei R. Yishmael: "sanctify him [the kohen]" in all manners of sanctification: [including] to open/begin first, to bless first, and to take the best portion first
The last example in the Gemara appears to be referring to having the Kohen take food or the like before non-Kohanim. How does this relate to being a דבר שבקדושה, to a matter of 'sanctification'? Does the Gemara simply mean that in all matters, allowing the Kohen to take precedence is an act of sanctifying him? That doesn't seem to be the most natural reading of the words.
DoubleAA reads the Gemara differently here, as though the Gemara is giving a list of four things, the first of them being כל דבר שבקדושה, however, I find this to be a highly implausible read. (Soncino and Artscroll translate the phrase as I have done above)
SO, how do you read this Gemara, and is it saying that a kohen should receive first pick in all matters, or only ones pertaining to holiness?