There are two separate issues that are getting mixed up here, as Tyler's comment above indicates:
- The point addressed in Mariano's answer: on the category of coherent sheaves, the functor of sections and $\mathrm{Hom}(\mathcal{O}_X,-)$ are isomorphic and thus have the same derived functor.
- The question of why the derived functor of sections on coherent sheaves is the same as when it is computed in the category of sheaves of abelian groups (on such a sheaf which happens to be coherent). This follows because every quasi-coherent sheaf has a quasi-coherent flasque resolution (Godement's canonical one), and you can check that flasque sheaves have trivial sheaf cohomology computed in either category.
Note, this is true for $\mathcal{O}_X$ any sheaf of rings on any topological space; in particular, for sheaves of abelian groups, the same statement is true for the sheaf $\mathbb{\underline{Z}}_X$ of locally constant $\mathbb{{Z}}$-valued functions.