In my answer [here][1] 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"][2] (from p. 88) by MacLane/Moerdijk, and an even more detailed and intuition-emphasizing one in Goldblatt's ["Topoi"][3], online viewable under the link.


  [1]: http://mathoverflow.net/questions/38966/what-is-sheaf-cohomology-intuitively/39081#39081
  [2]: http://books.google.com/books?id=SGwwDerbEowC&dq=sheaves+in+geometry+and+logic&printsec=frontcover&source=bn&hl=de&ei=aOeYTKPxJY2vOJbspeEO&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDAQ6AEwAw#v=onepage&q&f=false
  [3]: http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=Gold010