Szamuely's Galois Groups and Fundamental Groups might be what you're looking for. In particular, the beginning of Chapter 2 (where the discussion switches from field theory to fundamental groups) alludes to a statement like the one you give:
In the last section we saw that when studying extensions of some field it is plausible to conceive the base field as a point and a finite separable extension (or, more generally, a finite etale algebra) as a finite discrete set of points mapping to this base point. Galois theory then equips the situation with a continuous action of the absolute Galois group which leaves the base point fixed. It is natural to try to extend this situation by taking as a base not just a point but a more general topological space. The role of field extensions would then be played by certain con- tinuous surjections, called covers, whose fibres are finite (or, even more generally, arbitrary discrete) spaces. We shall see in this chapter that under some restrictions on the base space one can develop a topological analogue of the Galois theory of fields, the part of the absolute Galois group being taken by the fundamental group of the base space.
Edit: I notice that this book was discussed in another MO question here: http://mathoverflow.net/questions/546/galois-groups-vs-fundamental-groups/3686#3686 .
