Why does the following diagram commute?$\require{AMScd}$ \begin{CD} H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\ H^1(X,\mathscr{A}[\ell^n]) \times H^1(X,\mathscr{E}xt_X^1(\mathscr{A},\mu_{\ell^n})) @>>> H^2(X,\mu_{\ell^n}). \end{CD} ($\mathscr{A}/X$ is an Abelian scheme.) The vertical homomorphism $\delta$ on the left is induced by the Kummer sequence $$ 0 \to \mathscr{A}[\ell^n] \to \mathscr{A} \to \mathscr{A} \to 0. $$

The upper row is the Yoneda pairing.

The lower row is induced by the Weil pairing $$ \mathscr{A}[\ell^n] \times \mathscr{A}^\vee[\ell^n] \to \mu_{\ell^n} $$ using $\mathscr{E}xt^1(\mathscr{A},\mu_{\ell^n}) = \mathscr{H}om(\mathscr{A}[\ell^n],\mu_{\ell^n}) = \mathscr{A}^\vee[\ell^n]$ by the long exact Ext sequence for the Kummer sequence $0 \to \mathscr{A}[\ell^n] \to \mathscr{A} \to \mathscr{A} \to 0$: $$ 0 = \mathscr{H}om_X(\mathscr{A},\mu_{\ell^n}) \to \mathscr{H}om_X(\mathscr{A}[\ell^n],\mu_{\ell^n}) \to \mathscr{E}xt^1_X(\mathscr{A},\mu_{\ell^n}) \stackrel{0}{\to} \mathscr{E}xt^1_X(\mathscr{A},\mu_{\ell^n}), $$ the latter transition map being $0$ since $\ell^n$ kills $\mu_{\ell^n}$. (Edit 2: Or, better, $$ 0 = \mathscr{H}om_X(\mathscr{A},\mathbf{G}_m) \to \mathscr{E}xt^1_X(\mathscr{A},\mu_{\ell^n}) \to \mathscr{E}xt^1_X(\mathscr{A},\mathbf{G}_m) \stackrel{\ell^n}{\to} \mathscr{E}xt^1_X(\mathscr{A},\mathbf{G}_m) $$ and using $\mathscr{E}xt^1_X(\mathscr{A},\mathbf{G}_m) = \mathscr{A}^\vee$, as pointed out in the comments.)

Finally, for the middle vertical arrow: The local-to-global Ext spectral sequence $H^p(X,\mathscr{E}xt_X^q(\mathscr{A},\mu_{\ell^n})) \Rightarrow \mathrm{Ext}_X^{p+q}(\mathscr{A},\mu_{\ell^n})$ gives us an injection (an edge morphism, see also Natural morphism appearing in Grothendieck spectral sequence) $$ H^1(X,\mathscr{E}xt_X^1(\mathscr{A},\mu_{\ell^n})) \hookrightarrow \mathrm{Ext}_X^2(\mathscr{A},\mu_{\ell^n}) $$ since $\mathscr{H}om_X(\mathscr{A},\mu_{\ell^n}) = 0$.

Edit: Differently stated, why does \begin{CD} \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> \mathrm{Hom}(H^0(X,\mathscr{A}), H^2(X,\mu_{\ell^n})) \\ @AAA @A\delta^*AA \\ H^1(X,\mathscr{E}xt_X^1(\mathscr{A},\mu_{\ell^n})) @>>> \mathrm{Hom}(H^1(X,\mathscr{A}[\ell^n]), H^2(X,\mu_{\ell^n})) \end{CD} commute? (the left vertical arrow a edge map of the spectral sequence and the right vertical arrow $\delta^*$ with $\delta$ the connecting map of the Kummer sequence)