Here are the steps to transfer your backup from one device to another:
• Make a backup (iOS/Android: tap on Settings/Backup (or in the old versions, More/Backup; on Mac OS: File > Backup) 
• Save it to the cloud (Dropbox, Box, Files, GoogleDrive) or email it to yourself (or Apple only users use Airdrop*)
To import the whole backup – songs and playlists: (Requires iReal Pro 5.0.4 or higher)   
iOS: In Dropbox tap on the BACKUP.html file, on the top right, tap on Share, Export, Open in and select Import with iReal Pro.  In Mail tap and hold the BACKUP.html file and choose Copy to iReal Pro.
OS X: drag and drop the BACKUP.html file over the iReal Pro app icon in the dock of your computer.
Android: From the cloud, open the BACKUP.html file with iReal Pro. From Email you have to save the file to disk then open it using a file manager app.
* If both devices are Apple (Mac or iOS) you can use Airdrop to transfer your backup