Here are the steps to backup and restore if you're using Outlook for Mac.
1. Once Outlook for Mac is open, click on "Tools", then click on "Export".
2. Outlook for Mac will ask you to select what you want to backup, on the next screen check the box "Outlook for Mac Data File (.olm)".
Select what you'd like to backup, example below. Then click the bottom right arrow to go to the next screen.
3. Outlook will ask you if you want to delete your items after you export them, select the button "No, do not delete items", then click on the button right arrow again to go to the next screen.
4. Type in the name that you want your backup to have and where on your Mac you'd like to save it to and then click the "Save" button.
5. If you haven't already, set up the new account, like if you're moving to a hosted exchange account and want your emails from your old box on your new account. then you start the process to do the restore.
6. Once you've setup your new profile in Outlook, click on Tools, then click on the "Import" link.
7. You'll then click the button "Outlook for Mac Data File (.olm)" and click the bottom right arrow to go to the next step.
8. Browse to the folder where you saved your backup to, highlight the backup file and then click the "Import" button.
9. Once the import has completed, you will find your importaed mail in the restored email under "Archived Mail" folder under "On My Computer".