This is a screenshot from Hatcher's Algebraic Topology.
I can't understand the last sentence. By definition $C_n(X)$ is the free $\mathbb Z$-module generated by all n-simplices. But, why should every singular map go to exactly exactly $A$ or $X-A$ -- what about maps whose images lie in $A$ and a bit outside? And if this were true, why bother with barycentric subdivisions?