Step 1: Login to the cPanel.
Step 2: On the main cPanel page for the account find and click the “Email Accounts” button.
Step 3: find the email address you’d like to export emails for. Click the “More” button and then find and click “Access Webmail” and select 'Horde' webmail interface.
Step 4: From Horde, right click ‘Inbox’ and select ‘Export’ from the list.
Step 5: You will now see a popup with export options, you can simply click ‘OK’ to get the exported emails in MBOX format. The MBOX file format is a fairly generic format used for backing up Emails. The MBOX format is essentially a plain text file and has a wide range of support with various Email clients.