In your situation, the Serre spectral sequence looks as follows

\[H_p(C,H_q(A,K))\Rightarrow H_{p+q}(B,K).\]

The left hand side is the $E^2$ term. There, $H_q(A,K)$ carries an action of $\pi_1(C)$ induced by the fibration, and $H_p(C,H_q(A,K))$ is the homology with local coefficients. Now, if $A$ has the homology of a point then then $H_q(A,K)=0$ for $q>0$ and $H_0(A,K)=K$ with the trivial action of $\pi_1(C)$ (any self-map of a connected space induces the identity on $H_0$). Therefore $E^2_{p,q}=0$ for $q>0$ and $E^2_{n,0}=H_n(C,K)$ is the ordinary homology. Moreover, the spectral sequence ends in the second step and the edge morphism $H_n(B,K)\twoheadrightarrow E^\infty_{n,0}\subset E^2_{n,0}=H_n(C,K)$, which is induced by the map $B\rightarrow C$, is an isomorphism.