We first prove this for $p = 0$. Injectivity follows from the definition of $\mathcal{F}|_Z$ and holds in general (for any subset of any topological space $X$). Next, suppose that $s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$ such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$ comes from $s_i \in \mathcal{F}(U_i)$. It follows that there exist opens $W_{ij} \subset U_i \cap U_j$ with $W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that $s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset} we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and such that $V_i \cap V_j \subset W_{ij}$. Hence we see that $s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the open neighbourhood $\bigcup V_i$ of $Z$. \medskip\noindent To finish the proof, it suffices to show that if $\mathcal{I}$ is an injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$ for $p > 0$. This follows using short exact sequences and dimension shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$. By Lemma \ref{lemma-cech-Hausdorff-quasi-compact} the element $\overline{\xi}$ comes from $\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$ for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$. Say $\overline{\xi}$ is the image of the class of a cocycle $\xi = (\xi_{i_0 \ldots i_p})$ in $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$. \medskip\noindent Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf defined by the rule $$ \mathcal{I}'(V) = \{s \in \mathcal{I}|_Z(V) \mid \exists (U, t),\ U \subset X\text{ open}, \ t \in \mathcal{I}(U),\ V = Z \cap U,\ s = t|_{Z \cap U} \} $$ Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$. Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$ such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is a section of $\mathcal{I}'$. Applying Topology, Lemma \ref{topology-lemma-refine-covering} we may after refining $\mathcal{V}$ assume that each $\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$. \medskip\noindent Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$. Since $\mathcal{I}$ is flasque (Lemma \ref{lemma-injective-flasque}) and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$ for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$ which restricts to $\xi_{i_0 \ldots i_p}$ on $V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$. (This appeal to injectives being flasque can be avoided by an additional application of Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}.) Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain for the open covering $U = \bigcup U_i$. Since $\text{d}(\xi) = 0$ we see that the sections $\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial remarks of the proof, there exists open subsets $W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$ with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$ such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$. By Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset} we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$ and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$. Then $s' = (s'_{i_0 \ldots i_p})$ with $s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$ is a cocycle for $\mathcal{I}$ for the open covering $U' = \bigcup U'_i$ of an open neighbourhood of $Z$. Since $\mathcal{I}$ has trivial higher {\v C}ech cohomology groups (Lemma \ref{lemma-injective-trivial-cech}) we conclude that $s'$ is a coboundary. It follows that the image of $\xi$ in the {\v C}ech complex for the open covering $Z = \bigcup Z \cap U'_i$ is a coboundary and we are done.