You actually don't need all the machinery behind sheaves to understand this! The general concept is that of a terminal object in a category. In the category of sets, the terminal objects are the singletons: given a set S, there is only one function f:S -> {a}... send everything in S to a! In the category of topological spaces, the terminal objects are the single-point spaces.
That was my first thought as well, but the author directly linked this notion to sheaves and also was talking about mapping from points.
I guess this is because the structure sheaf of the spectrum of a field is a constant sheaf.