The isomorphism also holds for relative cohomology of sheaves on arbitrary topological spaces. For a proof, see for example Cor. 1.9 in Hartshorne's Local cohomology (LNM 41). It is quite elementary and self-contained. For some geometric intuition for relative cohomology you may consult texts on algebraic topology (for example Hatcher's textbook), because it coincides with relative singular cohomology in the following sense: If $(X,A)$ is a relative CW-complex and $G$ is a constant sheaf on $X$, there is a canonical isomorphism $H^i_A(X,G) \cong H^i_{\mathrm{sing}}(X,X \setminus A,G)$.
If you are not satisfied with the proofs for the isomorphism you ask for: What do you mean by explicitly? In which form do you want to represent the elements in both cohomology groups?