On several instances, Hartshorne states that certain questions are "local". What is the reasoning behind it? Some instances in the text by Hartshorne, algebraic geometry, occur on p. 114 f., proof of proposition 5.7, or proof of proposition 5.8.
In section 5 chapter 2, quasi-coherent sheaves are discussed. Quasi-coherent sheaves deal with $\mathcal{O}_X$-modules, which on some open covers of $X$ are isomorphic to a sheaf associated to a $A_i$-module for some appropriately chosen ring $A_i$. The elements of the open covers each equal the spectrum of a ring $A_i$.
Proposition 5.7 deals with a scheme $X$, which is not necessarily affine, and states that kernels, cokernels and images of morphisms between quasi-coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$ are quasi-coherent. Hartshorne simply states in the proof that by localness of the question, affineness can be assumed. I interpret it in the way that quasi-coherence requires that the restriction of a sheaf of modules to each element of some open covering is isomorphic to $\widetilde{M_{i}}$ where $M_{i}$ is a module over a ring $A_{i}$ with $\text{Spec}(A_{i})=U_{i}$.
An interpretation of the Hartshorne's explanation is that the property in question can be checked on the stalks. By proposition 1.1., p. 63 a morphism of sheaves on $X$ is an isomorphism if and only if the induced map on the stalks at $P$ is an isomorphism for each element $P$ of the topological space $X$. Certainly for an affine structure, the stalk of a sheaf of $A_i$-modules at $\mathfrak{p}$ is isomorphic to the localisation $(\widetilde{M_i})_{\mathfrak{p}}$, where $M_i\cong \Gamma(\text{Spec}(A_i),\widetilde{M_i})$ (proposition 5.1 (b) and (d)). An alternative interpretation might be that localness refers to the open covers $\{U_i\}_i$, but I am sightly doubtful about the latter interpretation.
In part (a) of proposition 5.8, the assumption of quasi-coherence of $\mathcal{G}$ is a local property on $X$ and $Y$. Also the claim regarding $f^{*} \mathcal{G}$ relates to quasi-coherence. I wonder why in proposition 5.8 part (c) it is claimed in the proof that the question is local on $Y$ only.
Especially part (c) of proposition 5.8 is unclear to me, but the other occurences of the approach "the question is local, so affineness can be assumed", remain unclear to me, as well.
Best Answer
Saying that a property $\mathcal{P}$ on a scheme is local means that $X$ has the property $\mathcal{P}$ if only if there is some open cover of $X=\bigcup_{i\in I}U_i$ such that $U_i$ has $\mathcal{P}$.
For instance, a sheaf $F$ on $X$ is quasi-coherent if and only if there is a covering $X=\bigcup_{i\in I} U_i$ such that $F|_{U_i}$ is quasi-coherent for each $i\in I$. So, if we want to show that some sheaf is quasi-coherent, we only need to check it on some open cover. This open cover can be one of your choosing, and if taking an affine cover makes the problem easier, then you can choose an affine open cover (this is what Hartshorne does).
Now, let us look at Propositon 5.7 specifically. Let $\phi:F\rightarrow G$ be a morphism of sheaves on our scheme $X$. Hartshorne want to show that $\operatorname{ker}(\phi),\operatorname{coker}(\phi)$ and $\operatorname{img}(\phi)$ are also quasi-coherent (these three denote the kernel, cokernel and image respectively). A fact Hartshorne is using to show this result is that for any open subset $U\subset X$: $$\operatorname{ker}(\phi|_{U})=\operatorname{ker}(\phi)|_{U},$$ $$\operatorname{coker}(\phi|_{U})=\operatorname{coker}(\phi)|_{U},$$ $$\operatorname{img}(\phi|_{U})=\operatorname{img}(\phi)|_{U}.$$ Now, to show that $\operatorname{ker}(\phi)$ is quasi-coherent, it is enough to consider an open cover $X=\bigcup_{i\in I}U_i$ and show that $\operatorname{ker}(\phi)|_{U_i}$ is quasi-coherent. Harshorne chooses this cover to be an affine cover because the problem is easier to solve on affine schemes. If for some $i\in I$, $F|_{U_i}\cong \widetilde{M_i}$, $G|_{U_i}\cong \widetilde{N_i}$ and $\phi|_{U_i}$ corresponds to $\Phi_i:M_i\rightarrow N_i$, then Hartshorne shows that $\operatorname{ker}(\phi)|_{U_i}\cong\operatorname{ker}(\phi|_{U_i})\cong\widetilde{\operatorname{ker}(\Phi_i)}$. This shows that $\operatorname{ker}(\phi)$ is quasi-coherent. Image and cokerenl are also done similarly.
Now, moving onto Proposition 5.8. I understand why that line in the proof of part (c) can be confusing. To clarify, quasi-coherence of a sheaf is local on whichever scheme it is defined over. The point that Hartshorne is trying to make is that $f_*\mathcal{F}$ is a sheaf on $Y$ and hence showing that it is quasi-coherent is local property on $Y$ (but not local on $X$ - explained below). That is why Hartshorne picks an open affine cover of $Y=\bigcup_{i\in I} Y_i$ whcih gives a cover (not necessarily affine) of $X=\bigcup X_{i}(=f^{-1}(Y_i))$, and then shows that result for $f|_{X_i}:X_i\rightarrow Y_i$. Again, as in the last Proposition, Hartshorne is in fact using the fact that $$(f|_{X_i*}\mathcal{F}|_{X_i})\cong (f_*\mathcal{F})|_{Y_i}.$$
I believe another doubt you had was why is part (a) local in both $X$ and $Y$, but that is not the case for $(c)$.
Part $(c)$ is not local on $X$ because if we take any affine open subset $U\subset X$ and $V\subset Y$ such that $f(U)\subset V$, then the following isomorphism is not true in general: $$(f|_{U*}\mathcal{F}|_{U})\cong (f_*\mathcal{F})|_{V}.$$
But for part (a), if we take any affine open subset $U\subset X$ and $V\subset Y$ such that $f(U)\subset V$, then we have the following isomorphism: $$(f|_{U}^*\mathcal{G}|_V)\cong (f^*\mathcal{G})|_{U}.$$ Hence, for this result, it is local in both $X$ and $Y$.
I hope this clarifies any doubt you had.