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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 .

2 added 1126 characters in body

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.