Note that the variance expression in the question is an approximation. Hedges (1981) derived the large sample variance of $d$ and approximation in a general setting (i.e. multiple experiments/studies), and my answer pretty much walks through the derivations in the paper.
First, the assumptions we'll utilize are the following:
Let's assume we have two independent treatment groups, $T$ (treatment) and $C$ (control). Let $Y_{Ti}$ and $Y_{Cj}$ be the scores/responses/whatever from subject $i$ in group $T$ and subject $j$ in group $C$, respectively.
We assume the responses are normally distributed and the treatment and control groups share a common variance, i.e.
\begin{align*}
Y_{Ti} &\sim N(\mu_T, \sigma^2), \quad i = 1, \dots n_T \\
Y_{Cj} &\sim N(\mu_C, \sigma^2), \quad j = 1, \dots n_C
\end{align*}
The effect size we're interested in estimating in each study is $\delta = \frac{\mu_T - \mu_C}{\sigma}$. The estimator of the effect size we'll use is
\begin{equation*}
d = \frac{\bar{Y}_T - \bar{Y}_C}{\sqrt{\frac{(n_T - 1)S_T^2 + (n_C - 1)S_C^2}{n_T + n_C - 2}}}
\end{equation*}
where $S_k^2$ is the unbiased sample variance for group $k$.
Let's consider the large-sample properties of $d$.
First, note that:
\begin{equation*}
\bar{Y}_T - \bar{Y}_C \sim N \Bigg( \mu_T - \mu_C, \,\sigma^2\frac{n_T + n_C}{n_T n_C} \Bigg)
\end{equation*}
and (being loose with my notation):
\begin{equation}
\frac{(n_T - 1)S_T^{2}}{\sigma^2(n_T + n_C - 2)} = \frac{1}{n_T + n_C - 2}\frac{(n_T - 1)S_T^{2}}{\sigma^2} \sim \frac{1}{n_T + n_C- 2}\chi_{n_T - 1}^2 \tag{1}
\end{equation}
and
\begin{equation}
\frac{(n_C - 1)S_C^{2}}{\sigma^2(n_T + n_C - 2)} = \frac{1}{n_T + n_C - 2}\frac{(n_C - 1)S_C^{2}}{\sigma^2} \sim \frac{1}{n_T + n_C- 2}\chi_{n_C - 1}^2 \tag{2}
\end{equation}
Equations (1) and (2) lead to the fact that (again, being loose with my notation):
\begin{equation*}
\frac{1}{\sigma^2}\frac{(n_T - 1)S_T^{2} + (n_C - 1)S_C^{2}}{n_T + n_C - 2} \sim \frac{1}{n_T + n_C - 2}\chi_{n_T + n_C - 2}^2
\end{equation*}
Now, some clever algebra:
\begin{align*}
d &= \frac{\bar{Y}_T - \bar{Y}_C}{\sqrt{\frac{(n_T - 1)S_T^2 + (n_C - 1)S_C^2}{n_T + n_C - 2}}} \\\\
&= \frac{\left(\sigma\sqrt{\frac{n_T + n_C}{n_T n_C}}\right)^{-1}(\bar{Y}_T - \bar{Y}_C)}{\left(\sigma\sqrt{\frac{n_T + n_C}{n_T n_C}}\right)^{-1}\sqrt{\frac{(n_T - 1)S_T^2 + (n_C - 1)S_C^2}{n_T + n_C - 2}}} \\\\
&= \frac{\frac{(\bar{Y}_T - \bar{Y}_C) - (\mu_T - \mu_C)}{\sigma\sqrt{\frac{n_T + n_C}{n_T n_C}}} + \frac{\mu_T - \mu_C}{\sigma\sqrt{\frac{n_T + n_C}{n_T n_C}}}}{\left(\sqrt{\frac{n_T + n_C}{n_T n_C}}\right)^{-1}\sqrt{\frac{(n_T - 1)S_T^2 + (n_C - 1)S_C^2}{\sigma^2(n_T + n_C - 2)}}} \\\\
&= \sqrt{\frac{n_T + n_C}{n_T n_C}}\left(\frac{\theta + \delta\sqrt{\frac{n_T n_C}{n_T + n_C}}}{\sqrt{\frac{V}{\nu}}}\right)
\end{align*}
where $\theta \sim N(0,1)$, $V \sim \chi^2_{\nu}$, and $\nu = n_T+n_C-2$. Thus, $d$ is $\sqrt{\frac{n_T + n_C}{n_T n_C}}$ times a variable which follows a non-central t-distribution with $n_T + n_C - 2$ degrees of freedom and non-centrality parameter of $\delta\sqrt{\frac{n_T n_C}{n_T + n_C}}$.
Using the moment properties of the non-central $t$ distribution, it follows that:
\begin{equation*}
\mathrm{Var}(d) = \frac{(n_T + n_C - 2)}{(n_T + n_C - 4)}\frac{(n_T + n_C)}{n_T n_C}(1+ \delta^2\frac{n_T n_C}{n_T + n_C}) - \frac{\delta^2}{b^2} \tag{3}
\end{equation*}
where
\begin{equation*}
b = \frac{\Gamma\left(\frac{n_T + n_C - 2}{2}\right)}{\sqrt{\frac{n_T+n_C-2}{2}}\Gamma\left(\frac{n_T+n_C-3}{2}\right)} \approx 1 - \frac{3}{4(n_T+n_C-2)-1}
\end{equation*}
So Equation (3) provides the exact large sample variance. Note that an unbiased estimator for $\delta$ is $b d$, with variance:
\begin{equation*}
\mathrm{Var}(bd) = b^2\frac{(n_T + n_C - 2)}{(n_T + n_C - 4)}\frac{(n_T + n_C)}{n_T n_C}(1+ \delta^2\frac{n_T n_C}{n_T + n_C}) - \delta^2
\end{equation*}
For large degrees of freedom (i.e. large $n_T+n_C-2$), the variance of a non-central $t$ variate with $\nu$ degrees of freedom and non-centrality parameter $p$ can be approximated by $1 + \frac{p^2}{2\nu}$ (Johnson, Kotz, Balakrishnan, 1995). Thus, we have:
\begin{align*}
\mathrm{Var}(d) &\approx \frac{n_T + n_C}{n_T n_C}\left(1 + \frac{\delta^2\left(\frac{n_T n_C}{n_T + n_C}\right)}{2(n_T+n_C-2)}\right)
\\\\
&= \frac{n_T + n_C}{n_T n_C} + \frac{\delta^2}{2(n_T+n_C-2)}
\end{align*}
Plug in our estimator for $\delta$ and we're done.
