$C^{n}(X,A;G) $can be viewed as functions from singular $n$ simplices in $X$ to $G$ that vanish on simplices in $A$? Hatcher says that for a pair of spaces $(X,A)$ and an abelian group $G$, we can view the relative cochain group $C^{n}(X,A,G)$ as functions from singular $n$-simplices in $X$ to $G$ that vanish on simplices in $A$.
He says that this is because the basis for $C_{n}(X)$ are the simplices in $X$ and we can write this as the disjoint union of simplices with imiages contained in $A$ and simplices with images not contained in $A.
I've been pondering this for awhile and don't quite see how his original statement follows. 
 A: It's best to consider the simplicial model of things when thinking about these things. If $X$ is a simplicial (or delta-, as Hatcher uses) complex, then $C_n(X)$ is the free $\Bbb Z$-module generated by the $n$-simplices in $X$. $C^n(X)$ is the dual vector space $\hom(C_n(X), \Bbb Z)$, elements of which are $\Bbb Z$-linear functionals on $C_n(X)$ - that in turn can be thought as an assignment of an integer to each $n$-simplex in $X$. So that's a "piece-wise linear function" on $X$ which is constant over the $n$-simplices.
The relative chain groups $C_n(X, A)$ are defined as $C_n(X)/C_n(A)$ for some subcomplex $A \subset X$. The elements of these are "chains in $X$ modulo chains in $A$". The dual group $C^n(X, A)$ consists of $\Bbb Z$-linear functionals $C_n(X, A) \to \Bbb Z$. These are the same as functionals $C_n(X)/C_n(A) \to \Bbb Z$, or by first isomorphism theorem, functionals $C_n(X) \to \Bbb Z$ which contains $C_n(A)$ in the kernel. By our previous analogy, that's a "piecewise linear function" on $X$, linear on the simplices of $X$, but vanishing on the simplices of the subcomplex $A \subset X$.
This analogy extends to singular cochain groups, but it's harder to explicitly describe the singular cochains as a function on $X$ then, as the space of singular simplices on a topological space is generally very huge.
