There are two separate issues that are getting mixed up here, as Tyler's comment above indicates:

1. 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.
2. 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.

There are two separate issues that are getting mixed up here, as Tyler's comment above indicates:

1. 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.
2. 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 (this argument is in the relevant section of the Stacks Project: http://stacks.math.columbia.edu/tag/09SV).

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.