[bash] What does if [ $? -eq 0 ] mean for shell scripts?

It is an extremely overused way to check for the success/failure of a command. Typically, the code snippet you give would be refactored as:

if grep -e ERROR ${LOG_DIR_PATH}/${LOG_NAME} > /dev/null; then
   ...
fi

(Although you can use 'grep -q' in some instances instead of redirecting to /dev/null, doing so is not portable. Many implementations of grep do not support the -q option, so your script may fail if you use it.)