As explained in How it works, mu4e communicates with its backend
(mu server) by sending commands and receiving responses (s-expressions).
For debugging purposes, it can be very useful to see this data. For this reason,
mu4e can log all these messages. Note that the ‘protocol’ is documented
to some extent in the mu-server manpage.
You can enable (and disable) logging with M-x mu4e-toggle-logging. The
log-buffer is called *mu4e-log*, and in the The main view, The headers view and The message view, there’s a keybinding $ that takes you there.
You can quit it by pressing q.
Logging can be a bit resource-intensive, so you may not want to leave it on all the time.
mu itself keeps a log as well; depending on your system that could be in
your systemd journal, syslog or in <MUHOME>/mu.log, on Unix
typically ~/.cache/mu/mu.log. To run the server in debug-mode, set
mu4e-mu-debug to t (before starting the server).