I have not been able to restore files from the backups that I have made with Apple Backup. When I attempt to restore, using Backup, the progress dialog appears to show the restore working correctly, but the resulting folder doesn't contain any files.
I found this comment on this post that describes a solution. The backup files created by Apple Backup contain disk images with the files being backuped. It is easy to get access to the files that you want to recover.
First use Backup to find the backup file that contains the file that you want to restore (if you are using incremental backups). Then run the following command to mount the disk image:
hdiutil mount -readonly BackupFile.IncrementalBackup/Contents/Contents/Backup.dmg