I'd like to add a quick little explanation to Georges's already sufficient answer (I'm sure this explanation is in Baez's post, but it can be said in a few lines here).
One of the morals of the famous Eckmann-Hilton lemma is that an abelian group is an abelian group object in the category of groups. Now, the classifying space functor $B: Grp \to Top$ preserves products (here $Top$ should be a convenient category of spaces like compactly generated weak Hausdorff spaces; see nLab). But product-preserving functors take algebraic gizmos (like abelian group objects) to algebraic gizmos. So $B$ takes abelian groups $G$ = abelian group objects in $Grp$ to abelian group objects in $Top$, i.e., topological abelian groups. And since we have a universal covering fibration $G \to EG \to BG$, we get $\pi_1(BG) \cong G$.
A refinement of this observation shows that we have a classifying space functor
$$B: TopAb \to TopAb$$
which can be used to realize topological abelian groups as infinite loop spaces.