I'm confused about Hatcher's Theorem 3.5 on page 203. The theorem states:

$H^n(X;G) \approx$ ker$d_n/$ Im$d_{n-1}$. Furthermore, the cellular cochain complex $\{H^{n}(X^n,X^{n-1};G),d_n\}$ is isomorphic to the dual of the cellular chain complex, obtained by applying Hom$(-,G)$.

I am particularly confused what it means for the cellular cochain complex to be isomorphich to the dual of the cellular chain complex.

Well, there is the cohomology complex with maps $$\delta:H^i(X^{i},X^{i-1}) \to H^{i+1}(X^{i+1},X^i)$$ and there is also the complex with maps $$d^*:\mathrm{Hom}(H_i(X),G) \to \mathrm{Hom}(H_{i+1}(X),G)$$ where $d^*(f)=f \circ d$.
To show an isomorphism, you want to exhibit a family of maps $$f_i:H^i(X^{i},X^{i-1}) \to \mathrm{Hom}(H_i(X),G)$$ where $f_{i+1} \circ \delta=d^* f_{i}$ (so that it commutes with the maps.)