Why does the distance between two members of a metric space map to a real number? Why does the distance between two members of a metric space map to a real number? The set of real numbers is itself (along with the Euclidean distance function) a metric space - in which case, real numbers are used to describe a property of real numbers (themselves).  How is that possible? (How can you use something to describe itself?)﻿  
 A: Requiring the distance function on the reals to be real-valued is no more circular than requiring that the sum of two reals be real.

That we use the real numbers for distance reflects the typical interests of geometers; prior examples that the notion of metric space was meant to generalize had real distances, and the kinds of spaces many people like to study are suitably measured by real distances. 
There's even a prior reason why the reals make a good target: they are the unique linearly ordered, complete, Archimedean abelian group. (e.g. see Wikipedia)
Specialized applications might prefer a different definition of metric; e.g. see Wikipedia again.
A: Real numbers (or the real line) provide a prototype for this concept.
Possibly you read an axiomatic definition of a metric space first where the author shows that real numbers is one such good example. This approach caused your confusion.
It is these properties of real numbers and the function $x-y$ there that is abstracted to give the definition of a metric space.
All definitions are abstractions of one well-understood example. SO nothing circular there. It is like the Statement: The set of natural numbers is countably infinite set. Proof: Take the identity function.
The countable sets are modelled on the set of natural numbers. 
A: You are right that the real numbers must be defined before metric spaces are defined since a the definition of a  metric space uses the real numbers.
However just because the real numbers happen to be a metric space does not mean that metric spaces were used in the definition of the real numbers. You define the real numbers (without any reference to the concept of a metric space), then define a metric space, then show that the real numbers are a metric space. No circular reasoning.
EDIT As a commenter noted, the usual metric applied to the rationals takes values in the rationals and thus doesn't require the reals to be defined to use... just modify the definition of the metric space so that the distance is rational-valued and everything stays in place. As was perhaps part of the confusion, the concept of distance is used in the definition of a Cauchy sequence, which is part of one possible definition of real numbers as equivalence classes of Cauchy sequences of rationals. You don't need to go through the full formalism of defining a (rational or ordered field - valued) metric space in order to use the notion of distance on the rationals, but you could certainly consistently bring it in here. And distance is an indispensable concept in this approach, so in a certain sense it's already there.
