This is just an elaboration on PVALs comment.
Given a short exact sequence $0 \to A \to B \to C \to 0$ of chain complexes, there is an induced long exact sequence in homology. This is constructed out what is known as the snake lemma: https://en.wikipedia.org/wiki/Snake_lemma
If you look in the construction of the boundary map, you will see that a key step is to use the differential of $B$. This can be interpreted topologically.
First lets just recall the construction of the relative chain complex short exact sequence. The idea is that the inclusion $A \to X$ induces an inclusion of (singular) chain complexes, and one can formally take the quotient to get the relative chain complex. (The quotient is also free, since a simplex is killed in the quotient iff it lied entirely in $A$, so we create no torsion after this quotient.)
Now we start with some representative $\alpha$ of an element in the relative homology group $H_n(X,A)$, which is some chain in $X$ whose boundary lies in $A$. $\partial \alpha$ is some chain lying in $A$, and since it was a boundary of a chain in $X$, we know that it is a cycle. But it may not be a boundary of some chain lying in $A$, hence could be some interesting homology class in $H_{n-1}(A)$.
It is useful to imagine the disc $D = X$ with boundary $S^1 = A$. The disc is a relative homology class in $H_2(X,A)$. It's boundary is a homology class in $H_1(A)$.
The long exact sequence asserts that in general a complete set of representatives for the homology classes in the kernel of the map $H_{n-1}(A) \to H_{n-1}(X)$ are obtained this way. But this really makes sense, since some $\beta$ being in the kernel of that map means that there appeared some chain $\alpha$ in $X$ filling in the holes of $\beta$, i.e. it became a boundary, or symbolically $\partial(\alpha) = \beta$. But now $\alpha$ is a chain in $C_n(X)$ whose boundary lies in $A$, hence it represents some relative cycle in $H_n(X,A)$, and its image under the map described before is exactly $\beta$.
Hope that is helpful.