Let's consider
$$
\begin{align}
\left({\phi}^{*},{\theta}^{*}\right) &\in
\operatorname*{arg\,max}_{\phi,\,\theta} \, \operatorname{ELBO}\left(\phi,\theta\right) \\&=
\operatorname*{arg\,min}_{\phi,\,\theta} \, \left\{-\operatorname{ELBO}\left(\phi,\theta\right)\right\} \\&=
\operatorname*{arg\,min}_{\phi,\,\theta} \, \left\{\operatorname{KL}\left(q_\phi\left(z\right)\|\ p_\theta\left(z|x\right)\right) - \ln\left(p_\theta\left(x\right)\right)\right\}
\end{align}
$$
for optimal parameters ${\theta}^{*}$ and ${\phi}^{*}$ with an asscociated optimal variational density $q_{\phi^*}\!\left(z\right)$.
In many Bayesian settings, where $\theta \equiv z$ is considered as random vector, an approximation to the posterior density $p_\theta\left(z|x\right) \equiv p\left(\theta|x\right)$ is sought and $p_\theta\left(x,z\right) \equiv p\left(x,\theta\right)$ is the product of the prior density $p_\theta\left(z\right) \equiv p\left(\theta\right)$ and the density of the observation model/likelihood $p_\theta\left(x|z\right) \equiv p\left(x|\theta\right)$. In this case, the ELBO and KL-divergence become independent of $\theta$ and maximizing $\operatorname{ELBO}\left(\phi\right)$ w.r.t. $\phi$ is equivalent to minimizing the KL-divergence w.r.t. $\phi$ since $p_\theta\left(x\right)$ is by assumption constant w.r.t. $\phi$. The KL-divergence is a common measure of discrepancy between two probabilit densities (but not a metric) which is non-negative and attains zero iff they coincide almost everywhere. One idea could thus be to find an "optimal" approximation $q_{\phi^*}\!\left(\theta\right)$ to the posterior density by minimizing the KL-divergence between $q_\phi\left(\theta\right)$ and $p\left(\theta|x\right)$ w.r.t. $\phi$. However, minimizing the KL-divergence directly is assumed to be inadmissible as it depends on the true posterior density, which is unknown or intractable (otherwise there would be no need to find an approximation to it in the first place). But in the evidence lower bound $\operatorname{ELBO}\left(\phi\right) \equiv \int \ln\left(p\left(x,\theta\right)/q_\phi\left(\theta\right)\right)q_\phi\left(\theta\right)\mathrm d \theta$, we are no longer dealing with $p\left(\theta|x\right)$ but with the joint density $p\left(x,\theta\right)$ which is assumed to have a known functional form.
In different settings, when dealing with models where the likelihood $p_\theta\left(x|z\right)$ is conditioned on a latent vector $z$, the "ELBO" is a log-likelihood lower bound. Now, by maximizing this lower bound, given by $\operatorname{ELBO}\left(\phi,\theta\right) \equiv \int \ln\left(p_\theta\left(x,z\right)/q_\phi\left(z\right)\right)q_\phi\left(z\right)\mathrm d z$, over $\phi$ and $\theta$, the goal is to both minimize the KL-divergence between $q_\phi\left(z\right)$ and $p_\theta\left(z|x\right)$ and maximize the approximate log-likelihood. Here, ${\theta}^{*}$ is a variational approximation to the MLE.
Reference
Ormerod, J. T., & Wand, M. P. (2010). Explaining variational approximations. The American Statistician, 64(2), 140-153.
