This nLab page uses the fact that from a short exact sequence $$ 0\rightarrow K \rightarrow L \rightarrow M \rightarrow 0 $$ of (topological, possibly discrete) Abelian groups, we get a long homotopy fiber sequence $$ \cdots B^{j-1}M \rightarrow B^j K \rightarrow B^j L \rightarrow B^j M \rightarrow B^{j+1}K \rightarrow \cdots $$ of iterated classifying spaces. (Here we use the fact that the classifying space of a topological Abelian group should give another.)
Could anyone suggest a reference that describes the details of the construction of the above long sequence?