One example would be a map induced by a morphism $f: X \to Y$ in the long homology sequence.
E.g. suppose the top row is a cohomology of pair $(X, A)$ and the bottom row is the cohomology of pair $(Y, B)$. Then the theorem says that the $H^n(X, A)$ can be squeezed between the $n$-th and $(n-1)$-th cohomology of $X$ and $A$, because any morphism inducing isomorphism on those extends to $H^n(X, A)$.
