Your other host should be able to help with how to restore the files on their server but if you download a backup created within your cPanel control panel, you can either extract the backup file on your pc and upload the files via ftp, or upload the archive to your new server and extract it. If they run cPanel, you should be able to restore /home , alias and mysql backups using cPanel on the new server.
The files are in the following format, there may be others too, these are just the basic ones.
public_html - your web document root directory
etc - contains mail, added ftp account logins, quotas and passwords (This may be no good unless your new host uses cPanel)
.spamassassin - spamassassin config files
tmp - stats history and files
mail - mail boxes
.neomail - neomail files