Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Since every standard open $D(f)$ is quasi-compact we see that $X$ is a locally quasi-compact, i.e., every point has a fundamental system of quasi-compact neighbourhoods, see Topology, Definition \ref{topology-definition-locally-quasi-compact}. Hence by Modules, Lemma \ref{modules-lemma-quasi-coherent-module} for every prime $\mathfrak p \subset R$ corresponding to $x \in X$ there exists an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_U$ is isomorphic to the quasi-coherent sheaf associated to some $\mathcal{O}_X(U)$-module $M$. In other words, we get an open covering by $U$'s with this property. By Lemma \ref{lemma-standard-open} for example we can refine this covering to  a standard open covering. Thus we get a covering $\Spec(R) = \bigcup D(f_i)$ and $R_{f_i}$-modules $M_i$ and isomorphisms $\varphi_i : \mathcal{F}|_{D(f_i)} \to \mathcal{F}_{M_i}$ for some $R_{f_i}$-module $M_i$. On the overlaps we get isomorphisms $$ \xymatrix{ \mathcal{F}_{M_i}|_{D(f_if_j)} \ar[rr]^{\varphi_i^{-1}|_{D(f_if_j)}} & & \mathcal{F}|_{D(f_if_j)} \ar[rr]^{\varphi_j|_{D(f_if_j)}} & & \mathcal{F}_{M_j}|_{D(f_if_j)}. } $$ Let us denote these $\psi_{ij}$. It is clear that we have the cocycle condition $$ \psi_{jk}|_{D(f_if_jf_k)} \circ \psi_{ij}|_{D(f_if_jf_k)} = \psi_{ik}|_{D(f_if_jf_k)} $$ on triple overlaps. \medskip\noindent Recall that each of the open subspaces $D(f_i)$, $D(f_if_j)$, $D(f_if_jf_k)$ is an affine scheme. Hence the sheaves $\mathcal{F}_{M_i}$ are isomorphic to the sheaves $\widetilde M_i$ by Lemma \ref{lemma-compare-constructions} above. In particular we see that $\mathcal{F}_{M_i}(D(f_if_j)) = (M_i)_{f_j}$, etc. Also by Lemma \ref{lemma-compare-constructions} above we see that $\psi_{ij}$ corresponds to a unique $R_{f_if_j}$-module isomorphism $$ \psi_{ij} : (M_i)_{f_j} \longrightarrow (M_j)_{f_i} $$ namely, the effect of $\psi_{ij}$ on sections over $D(f_if_j)$. Moreover these then satisfy the cocycle condition that $$ \xymatrix{ (M_i)_{f_jf_k} \ar[rd]_{\psi_{ij}} \ar[rr]^{\psi_{ik}} & & (M_k)_{f_if_j} \\ & (M_j)_{f_if_k} \ar[ru]_{\psi_{jk}} } $$ commutes (for any triple $i, j, k$). \medskip\noindent Now Algebra, Lemma \ref{algebra-lemma-glue-modules} shows that there exist an $R$-module $M$ such that $M_i = M_{f_i}$ compatible with the morphisms $\psi_{ij}$. Consider $\mathcal{F}_M = \widetilde M$. At this point it is a formality to show that $\widetilde M$ is isomorphic to the quasi-coherent sheaf $\mathcal{F}$ we started out with. Namely, the sheaves $\mathcal{F}$ and $\widetilde M$ give rise to isomorphic sets of glueing data of sheaves of $\mathcal{O}_X$-modules with respect to the covering $X = \bigcup D(f_i)$, see Sheaves, Section \ref{sheaves-section-glueing-sheaves} and in particular Lemma \ref{sheaves-lemma-mapping-property-glue}. Explicitly, in the current situation, this boils down to the following argument: Let us construct an $R$-module map $$ M \longrightarrow \Gamma(X, \mathcal{F}). $$ Namely, given $m \in M$ we get $m_i = m/1 \in M_{f_i} = M_i$ by construction of $M$. By construction of $M_i$ this corresponds to a section $s_i \in \mathcal{F}(U_i)$. (Namely, $\varphi^{-1}_i(m_i)$.) We claim that $s_i|_{D(f_if_j)} = s_j|_{D(f_if_j)}$. This is true because, by construction of $M$, we have $\psi_{ij}(m_i) = m_j$, and by the construction of the $\psi_{ij}$. By the sheaf condition of $\mathcal{F}$ this collection of sections gives rise to a unique section $s$ of $\mathcal{F}$ over $X$. We leave it to the reader to show that $m \mapsto s$ is a $R$-module map. By Lemma \ref{lemma-compare-constructions} we obtain an associated $\mathcal{O}_X$-module map $$ \widetilde M \longrightarrow \mathcal{F}. $$ By construction this map reduces to the isomorphisms $\varphi_i^{-1}$ on each $D(f_i)$ and hence is an isomorphism.