1. If your CMS (myBB) has a "backup" function, you can use that. Download the backup, upload to new server. Your CMS's home site should have information on how to do it.
2. If installed via Softaculous, check the command panel in cPanel. It sometimes has a backup feature.
3. Lastly, you can get a copy of your data base files by using the "Export" feature in cPanel-->phpMyAdmin (it downloads automatically) and then you can Zip your files by going to cPanel-->FileManager, etc and then downloading the resulting .zip file. Import the database and upload/unzip the zip file to your new server.