The Administration Console allows you to see log messages (also available in a log file) and change the log settings.
While the first feature is useful only if you don't have access to the log files (inspecting log files with Unix command-line tools is definitely more powerful than doing the same with the AJAX-refreshed page), managing log settings is very useful because it doesn't require manual edits or server restarts. So, if you want to limit the priority level of log messages on-the-fly, or debug the behavior of a component, this is the right place to do so.
Tip
A verbose log level can slow down index operations, so it's better to check log settings before calling the /update request handler. For the same reason, remember that Solr logs all query requests at the INFO level. Depending on how many users your application has, this could lead to a huge amount of log messages.