Trying to understand free-energy equations in a Karl Friston neuroscience article
I'm not qualified to understand almost all of that paper, but, I might be able to give some intuitions from information theory that help you parse the paper.

|| denotes the Kullback-Leibler divergence. It measures an information gain between two distributions. I suppose you could say it indicates the information in the real distribution of data that a model fails to capture.

When you see "negative log" think "entropy".

In the first equation, think of it as "-ln(...) - -ln(...)". This may help think of it as the difference of entropies. Likewise in the second, read it as "D(...) + -ln(...)". This may help think of it as "plus entropy".

If you look at the divergence definition, you'll see it is defined as the log of the ratio of the PDFs. This may help connect it to logs and negative logs. Look at the definition that writes it as cross-entropy minus entropy. Then this is all a question of differences of entropies of things which may be clearer.