Mail accumulates until it causes a problem, then it causes several at once.
WHY IT MATTERS
Mail shares your hosting disk quota. A full account stops accepting uploads, stops delivering mail, and can make databases read-only.
ROUTINE HOUSEKEEPING
Empty Trash and Junk regularly. These are usually the largest folders. Delete large attachments once saved elsewhere. Sort by size in webmail to find them. Delete newsletters and notifications you have read.
ARCHIVING PROPERLY
Connect the mailbox in Thunderbird or Outlook, create a local folder, and move old mail into it. The mail is then on your computer rather than the server, and should be included in whatever backs that computer up.
Do this annually, keeping the current year and the previous one on the server.
WHAT NOT TO DO
Do not delete mail you may need for tax, legal or contractual reasons. Archive it instead. Do not rely on one person's laptop as the only copy of the business's correspondence.
IF SPACE IS SHORT NOW
cPanel > Email Accounts shows usage per mailbox. Start with the largest.