Making a backup with Mailbird is easy!
All you need to do is open your file explorer, and go to: C:\Users\"your user name"\AppData\Local and copy the Mailbird folder that is there. Mailbird application should be CLOSED at the time of copying.
Next, Install Mailbird in your new PC. After the installation is complete, go to the same location as the above, on your new PC and paste/overwrite the existing Mailbird folder with the back up that you've created from your old one.
And that's it! You can open and use Mailbird on your new PC with all your existing settings and email.