Homebrew running service shows error, how to debug this?
Look at the log file in /usr/local/var/log/elasticsearch.log.
TL;DR
If you look at the formula definition with:
brew edit elasticsearch@6

You will find the content of the plist file used for this service, which define the output log as:
#{var}/log/elasticsearch.log

According to the documentation:

#var ⇒ Object
The directory where the formula's variable files should be installed. This directory is not inside the HOMEBREW_CELLAR so it persists across upgrades.

According to the source code:
def var
  HOMEBREW_PREFIX/&quot;var&quot;
end

You can find the HOMEBREW_PREFIX with:
brew --prefix

It is most likely /usr/local. So you will find your log file in /usr/local/var/log/elasticsearch.log.