In my answer here I indicated how sheaf sections are sections of maps. This doesn't involve the structure map, but clearly motivates the name, if that's what you were looking for.
A more detailed account is in "Sheaves in Geometry and Logic" (from p. 88) by MacLane/Moerdijk, and an even more detailed and intuition-emphasizing one in Goldblatt's "Topoi", online viewable under the link.
