[proofplan]
We identify degree-zero Čech cochains with families of local sections on the members of the cover. The Čech cocycle condition says exactly that these local sections agree on all pairwise overlaps. By the sheaf gluing axiom, such compatible families are in canonical bijection with global sections of $\mathcal{F}$ on $X$. Since there are no nonzero $(-1)$-cochains in the Čech complex, degree-zero cohomology is the kernel of the first Čech differential, hence this bijection identifies $\check{H}^0(\mathcal{U}, \mathcal{F})$ with $\mathcal{F}(X)=\Gamma(X,\mathcal{F})$.
[/proofplan]
[step:Write degree-zero Čech cochains as families of local sections]
Let $X$ denote the underlying topological space, let $\mathcal{F}$ be a sheaf of abelian groups on $X$, and let $\mathcal{U}=(U_i)_{i \in I}$ be an open cover of $X$ indexed by a set $I$. For indices $i,j \in I$, define the open intersection $U_{ij}:=U_i \cap U_j$.
By definition of the Čech cochain complex for the cover $\mathcal{U}$, the degree-zero cochain group is
\begin{align*}
\check{C}^0(\mathcal{U},\mathcal{F})
=
\prod_{i \in I} \mathcal{F}(U_i).
\end{align*}
Thus an element $f \in \check{C}^0(\mathcal{U},\mathcal{F})$ is a family $f=(f_i)_{i \in I}$ with $f_i \in \mathcal{F}(U_i)$ for every $i \in I$.
[/step]
[step:Translate the cocycle condition into agreement on overlaps]
The Čech differential
\begin{align*}
\delta^0:\check{C}^0(\mathcal{U},\mathcal{F}) \to \check{C}^1(\mathcal{U},\mathcal{F})
\end{align*}
is defined on a family $f=(f_i)_{i \in I}$ by
\begin{align*}
(\delta^0 f)_{ij}
=
f_j|_{U_{ij}} - f_i|_{U_{ij}}
\end{align*}
for every pair of indices $i,j \in I$. Therefore $f \in \ker(\delta^0)$ if and only if
\begin{align*}
f_i|_{U_{ij}} = f_j|_{U_{ij}}
\end{align*}
for every $i,j \in I$. Hence $\ker(\delta^0)$ is precisely the set of compatible families of local sections on the cover $\mathcal{U}$.
[guided]
We now unpack what it means for a degree-zero cochain to be a cocycle. The map
\begin{align*}
\delta^0:\check{C}^0(\mathcal{U},\mathcal{F}) \to \check{C}^1(\mathcal{U},\mathcal{F})
\end{align*}
measures the failure of a family of local sections to agree on overlaps. Given $f=(f_i)_{i \in I}$ with $f_i \in \mathcal{F}(U_i)$, its image under $\delta^0$ has $(i,j)$-component
\begin{align*}
(\delta^0 f)_{ij}
=
f_j|_{U_{ij}} - f_i|_{U_{ij}},
\end{align*}
where $U_{ij}=U_i \cap U_j$. Thus $\delta^0 f=0$ exactly when every component is zero:
\begin{align*}
f_j|_{U_{ij}} - f_i|_{U_{ij}} = 0
\end{align*}
for all $i,j \in I$. Since $\mathcal{F}(U_{ij})$ is an abelian group, this is equivalent to
\begin{align*}
f_i|_{U_{ij}} = f_j|_{U_{ij}}.
\end{align*}
So the cocycle condition is not an additional mysterious cohomological condition: in degree zero it is exactly the sheaf-theoretic compatibility condition on pairwise intersections.
[/guided]
[/step]
[step:Use the sheaf gluing axiom to identify compatible families with global sections]
Define the restriction map
\begin{align*}
\rho:\mathcal{F}(X) &\to \ker(\delta^0) \\
s &\mapsto (s|_{U_i})_{i \in I}.
\end{align*}
For any $s \in \mathcal{F}(X)$ and any $i,j \in I$, the identity axiom for restrictions gives
\begin{align*}
(s|_{U_i})|_{U_{ij}} = s|_{U_{ij}} = (s|_{U_j})|_{U_{ij}},
\end{align*}
so $\rho(s) \in \ker(\delta^0)$. Conversely, if $f=(f_i)_{i \in I} \in \ker(\delta^0)$, then the previous step gives $f_i|_{U_{ij}}=f_j|_{U_{ij}}$ for all $i,j \in I$. Since $\mathcal{F}$ is a sheaf and $\mathcal{U}$ covers $X$, the gluing axiom gives a unique section $s \in \mathcal{F}(X)$ satisfying $s|_{U_i}=f_i$ for every $i \in I$. Hence $\rho$ is bijective.
Because restriction maps in a sheaf of abelian groups are group homomorphisms, $\rho$ is a group homomorphism. Therefore $\rho$ is a canonical isomorphism of abelian groups
\begin{align*}
\mathcal{F}(X) \cong \ker(\delta^0).
\end{align*}
[guided]
The bridge from Čech cocycles to global sections is the sheaf gluing axiom. First define the map from global sections to compatible families:
\begin{align*}
\rho:\mathcal{F}(X) &\to \ker(\delta^0) \\
s &\mapsto (s|_{U_i})_{i \in I}.
\end{align*}
This map is well-defined because restrictions are compatible with further restrictions. Indeed, for any $i,j \in I$,
\begin{align*}
(s|_{U_i})|_{U_{ij}} = s|_{U_{ij}} = (s|_{U_j})|_{U_{ij}}.
\end{align*}
Thus the family $(s|_{U_i})_{i \in I}$ satisfies the cocycle condition and lies in $\ker(\delta^0)$.
Now take an arbitrary cocycle $f=(f_i)_{i \in I}\in \ker(\delta^0)$. From the cocycle condition, we have
\begin{align*}
f_i|_{U_{ij}} = f_j|_{U_{ij}}
\end{align*}
for every $i,j \in I$. These are exactly the compatibility hypotheses required by the sheaf gluing axiom. Since $\mathcal{U}=(U_i)_{i \in I}$ covers $X$, the gluing axiom produces a section $s \in \mathcal{F}(X)$ such that
\begin{align*}
s|_{U_i}=f_i
\end{align*}
for every $i \in I$. The uniqueness clause in the sheaf axiom says that no other global section has these same restrictions to all $U_i$, so this construction is inverse to $\rho$. Therefore $\rho$ is a bijection.
Finally, because $\mathcal{F}$ is a sheaf of abelian groups, each restriction map $\mathcal{F}(X)\to \mathcal{F}(U_i)$ is a group homomorphism. Hence $\rho$ preserves addition componentwise, so the bijection is an isomorphism of abelian groups:
\begin{align*}
\mathcal{F}(X) \cong \ker(\delta^0).
\end{align*}
[/guided]
[/step]
[step:Quotient by zero coboundaries to obtain degree-zero Čech cohomology]
By definition,
\begin{align*}
\check{H}^0(\mathcal{U},\mathcal{F})
=
\ker(\delta^0)/\operatorname{im}(\delta^{-1}).
\end{align*}
The Čech complex has no degree $-1$ cochains, so $\check{C}^{-1}(\mathcal{U},\mathcal{F})=0$ and therefore $\operatorname{im}(\delta^{-1})=0$. Hence
\begin{align*}
\check{H}^0(\mathcal{U},\mathcal{F})
=
\ker(\delta^0).
\end{align*}
Combining this equality with the canonical isomorphism from the previous step gives
\begin{align*}
\check{H}^0(\mathcal{U},\mathcal{F})
\cong
\mathcal{F}(X).
\end{align*}
Finally, by definition of global section notation, $\Gamma(X,\mathcal{F})=\mathcal{F}(X)$. Therefore
\begin{align*}
\check{H}^0(\mathcal{U},\mathcal{F}) \cong \mathcal{F}(X)=\Gamma(X,\mathcal{F}),
\end{align*}
as required.
[/step]
