You stated you have a backup file: backup_201605202147.zip
How large is that file? If it is only 300kb then that is what existed when it was created. Delete this file if it is only 300kb as it is useless.
On the old machine , open eM Client and use the backup option to create another backup. It should be many megabytes in size if you old eM Client folder exceeds 600Mb.
Once confirming you have a valid backup, now transfer that new backup to you new computer.
On the new computer setup eM Client to create backups located at the place where you just transferred your new backup from the old computer.
Now select the Restore… option and it should restore from the backup you created.