[proofplan]
We compare the Čech complex of the fixed cover $\mathcal{U}$ with the global Čech cohomology of $X$ through the Leray spectral sequence for the cover. The $E_2$ page has entries obtained by taking Čech cohomology of $\mathcal{U}$ with coefficients in the presheaves $V \mapsto \check{H}^r(V,\mathcal{F}|_V)$. The Leray hypothesis makes every positive-degree coefficient group vanish on every finite intersection from $\mathcal{U}$, so all rows $r \geq 1$ are zero. The remaining row is exactly the Čech cohomology of $\mathcal{U}$ with coefficients in $\mathcal{F}$, and concentration on this row makes the edge morphism an isomorphism onto $\check{H}^q(X,\mathcal{F})$.
[/proofplan]
[step:Name the finite intersections and the coefficient presheaves used by the spectral sequence]
Write $\mathcal{U}=\{U_i\}_{i\in I}$, where $I$ is the index set of the cover. For every integer $p\geq 0$ and every ordered tuple $\mathbf{i}=(i_0,\ldots,i_p)\in I^{p+1}$, define the open set
\begin{align*}
U_{\mathbf{i}}:=U_{i_0}\cap\cdots\cap U_{i_p}\subseteq X.
\end{align*}
For every integer $r\geq 0$, define the Čech-cohomology presheaf
\begin{align*}
\mathcal{H}^r_{\mathcal{F}}:\operatorname{Open}(X)^{\mathrm{op}}&\to \mathbf{Ab}\\
V&\mapsto \check{H}^r(V,\mathcal{F}|_V),
\end{align*}
where $\operatorname{Open}(X)$ is the category of open subsets of $X$, $\mathbf{Ab}$ is the category of abelian groups, and the restriction morphisms are induced by restricting $\mathcal{F}|_V$ to smaller open subsets.
The hypothesis that $\mathcal{U}$ is a Leray cover for $\mathcal{F}$ means that, for every integer $r\geq 1$ and every ordered tuple $\mathbf{i}\in I^{p+1}$ with $p\geq 0$,
\begin{align*}
\mathcal{H}^r_{\mathcal{F}}(U_{\mathbf{i}})
=\check{H}^r(U_{\mathbf{i}},\mathcal{F}|_{U_{\mathbf{i}}})
=0.
\end{align*}
When $U_{\mathbf{i}}=\varnothing$, the same equality holds because the restriction of a sheaf of abelian groups to the empty open set has zero Čech cohomology in positive degree.
[/step]
[step:Apply the Leray spectral sequence for the cover $\mathcal{U}$]
By the Leray spectral sequence for Čech cohomology, applied to the sheaf of abelian groups $\mathcal{F}$ on $X$ and the open cover $\mathcal{U}$, there is a first-quadrant cohomological spectral sequence
\begin{align*}
E_2^{p,r}=\check{H}^p(\mathcal{U},\mathcal{H}^r_{\mathcal{F}})
\Longrightarrow \check{H}^{p+r}(X,\mathcal{F}),
\end{align*}
for integers $p,r\geq 0$. The same theorem identifies the edge morphism from the row $r=0$,
\begin{align*}
E_2^{q,0}\to \check{H}^q(X,\mathcal{F}),
\end{align*}
with the natural map from the Čech cohomology of the cover to global Čech cohomology, after the standard identification $\mathcal{H}^0_{\mathcal{F}}\cong\mathcal{F}$.
[/step]
[step:Show that all positive rows of the $E_2$ page vanish]
Let $r\geq 1$ be an integer. For every integer $p\geq 0$, the $p$-th Čech cochain group of $\mathcal{U}$ with coefficients in $\mathcal{H}^r_{\mathcal{F}}$ is
\begin{align*}
\check{C}^p(\mathcal{U},\mathcal{H}^r_{\mathcal{F}})
=\prod_{\mathbf{i}\in I^{p+1}}\mathcal{H}^r_{\mathcal{F}}(U_{\mathbf{i}}).
\end{align*}
Each factor in this product is zero by the Leray condition recorded above. Hence
\begin{align*}
\check{C}^p(\mathcal{U},\mathcal{H}^r_{\mathcal{F}})=0
\end{align*}
for every $p\geq 0$. The Čech differential on the zero cochain groups is the zero homomorphism, so
\begin{align*}
E_2^{p,r}
=\check{H}^p(\mathcal{U},\mathcal{H}^r_{\mathcal{F}})
=0
\end{align*}
for every $p\geq 0$ and every $r\geq 1$.
[/step]
[step:Identify the remaining row with the Čech cohomology of $\mathcal{U}$ with coefficients in $\mathcal{F}$]
For every open subset $V\subseteq X$, the canonical map
\begin{align*}
\eta_V:\mathcal{F}(V)&\to \check{H}^0(V,\mathcal{F}|_V)\\
s&\mapsto [s]
\end{align*}
is an isomorphism by the sheaf gluing axiom and the definition of zeroth Čech cohomology. These isomorphisms commute with restriction to smaller open subsets, so they define an isomorphism of presheaves
\begin{align*}
\eta:\mathcal{F}\xrightarrow{\sim}\mathcal{H}^0_{\mathcal{F}}.
\end{align*}
Applying Čech cohomology of the fixed cover $\mathcal{U}$ to this presheaf isomorphism gives
\begin{align*}
E_2^{p,0}
=\check{H}^p(\mathcal{U},\mathcal{H}^0_{\mathcal{F}})
\cong \check{H}^p(\mathcal{U},\mathcal{F})
\end{align*}
for every integer $p\geq 0$.
[/step]
[step:Use concentration on the zero row to identify the edge map as an isomorphism]
Fix an integer $n\geq 0$, and define
\begin{align*}
A^n:=\check{H}^n(X,\mathcal{F}).
\end{align*}
The spectral sequence is first-quadrant and satisfies $E_2^{p,r}=0$ for $r\geq 1$. Therefore every later page also satisfies
\begin{align*}
E_\infty^{p,r}=0
\end{align*}
for $r\geq 1$, while
\begin{align*}
E_\infty^{p,0}=E_2^{p,0}.
\end{align*}
Indeed, every differential into or out of the row $r=0$ either has source in a positive row already equal to zero or target in a negative row, which is absent in a first-quadrant spectral sequence.
Convergence gives a finite filtration of $A^n$ by subgroups
\begin{align*}
A^n=\Phi^0A^n\supseteq \Phi^1A^n\supseteq\cdots\supseteq \Phi^{n}A^n\supseteq \Phi^{n+1}A^n=0
\end{align*}
such that
\begin{align*}
\Phi^pA^n/\Phi^{p+1}A^n\cong E_\infty^{p,n-p}
\end{align*}
for $0\leq p\leq n$. Since $E_\infty^{p,n-p}=0$ unless $p=n$, this filtration has zero successive quotients except at $p=n$, and the edge morphism
\begin{align*}
E_2^{n,0}=E_\infty^{n,0}\to A^n
\end{align*}
is an isomorphism. Using the identification $E_2^{n,0}\cong\check{H}^n(\mathcal{U},\mathcal{F})$, this is precisely the natural map
\begin{align*}
\check{H}^n(\mathcal{U},\mathcal{F})\to \check{H}^n(X,\mathcal{F}).
\end{align*}
Thus the natural map is an isomorphism for every integer $n\geq 0$, which is the desired assertion.
[/step]
