Based on my experience, and confirmation
here:
- Reinstall VirtualBox
- Go to System Preferences -> Security & Privacy -> General
- Grant permission to "Oracle America, Inc" - note you may need to click on the padlock first
- Restart
You should be able to launch Virtual Box, but there will still be some extra permissions to grant when you launch your first VM. Given the reputation of VirtualBox, you are safe to grant what it needs.
This is all a side effect of Apple making it a little more challenging to install kernel level drivers and being over zealous about security and privacy.