trying to understand the connecting homomorphism between K theory groups

Question

the connecting homomorphism from $K_{1}(A/J)$ to $K_{0}(J)$ is defined by the composition $(j_{*})^{-1} k_{*}$ where $j$ is the inclusion of $J$ to the mapping cone $C_{\pi}$, which induce isomorphism between $K_{0}(J)$ and $K_{0}(C_{\pi})$. and $k$ is the inclusion of $S(A/J)$ in $C_{\pi}$ . I am trying to understand what does the connecting homomorphism exactly do, is there a geometric explanation? I am reading a book about c star algebras K theory and it mention no geometry counterpart.