What's going on in the equation of the variational lower bound? I don't really understand what this equation is saying or what the purpose of the ELBO is. How does it help us find the true posterior distribution?
From this document, as you found here, $X$ is an observed variable and $Z$ is a hidden variable; $p(X)$ is the density function of $X$. The posterior distribution of the hidden variables can then be written as follows using the Bayes’ Theorem:

$$p(Z|X) = \frac{p(X|Z)p(Z)}{p(X)} = \frac{p(X|Z)p(Z)}{\int_Zp(X,Z)}$$

Now base on what you post, if we denote that $L= \mathbb{E}_q [\log p(X, Z)] + H[Z]$ ($q(Z)$ is a distribution we use to approximation the true posterior distribution $p(Z|X)$ in VB and $H[Z] = -\mathbb{E}_q [\log q(Z)]$), then it is obvious that $L$ is a lower bound of the log probability of the observations.
As a result, if in some cases we want to maximize the marginal probability (the log probability of the observations), we can instead maximize its variational lower bound $L$. As a real example, you can follow the "Multiple Object Recognition with Visual Attention" example in the referenced document.

Moreover, the term $L$ will be presented in KL-divergence that will be used to measuring the similarity of two distributions. Be aware that there is progress on the bound in this paper (Fixing a Broken ELBO).