It is possible to export a MySQL backup for your MyBB installation using this functionality.
Under Database Backups you will find all backups for your MyBB database created via the backup functionality of MyBB that were stored locally on the server.
Under New Backup you will be able to manually create a backup. It can be stored on the server or downloaded locally to your computer.
Via the Optimize Database section you can select your MyBB database tables and using its automatic functionality analyze and optimize them for better performance. It is recommended to regularly run this optimization in order to keep the forum running as smoothly as possible.
The whole Logs section of Tools and Maintenance sole functionality is to provide logs for the activities that took place under your MyBB forum. You can purge old activities if you no longer need them.