If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong \tilde{H}_n(X;A)$$ if $X$ has the homotopy type of a CWcomplex. Can anyone point me to a proof of this fact?

This result appears in M. C. McCord, Classifying Spaces and Infinite Symmetric Products published in Transactions of the American Mathematical Society, Vol. 146, (Dec., 1969), pp. 273298. Also, this can be generalized to other generalized homology theories: given any spectrum $E$ one can build out of it a partial monoid, $A$, for which a variant of the construction you describe produces, given a space $X$, a new space whose homotopy groups are the reduced $E$homology of $X$. This is proved in Jacob Mostovoy's paper Partial monoids and DoldThom functors. 

