In case your previous hosting provider have not used cPanel and the transfer email accounts and messages between cPanel servers procedure is not applicable for you, but you still want to transfer your emails to SiteGround, please follow the steps below:
1) Create the email addresses in the cPanel of your SiteGround account.
2) Install a local email client on your computer (Outlook, Thunderbird, Mac Mail, etc.) and configure it to work with the settings of previous hosting provider (you can find those settings in your account there)
3) When configuring your email account, set it to work with IMAP
4) In the same mail client add and configure the same email account – (e.g. email@example.com) – but with SiteGround IMAP settings this time. You can find those in your cPanel -> Email Accounts -> More (button next the firstname.lastname@example.org email address) -> Configure Email Client.
5) Here comes the easy part – transfer your emails by dragging the content of each folder you have from the first email@example.com to the second (one with SiteGround configuration).
6) Once you complete the email transfer you can safely remove the first firstname.lastname@example.org and to continue using your mail account with SiteGround.