On-premises, this would allow a user account to logon to the network and access resources as would any other user (workstations, file shares, LDAP integrated services, federated services, etc.) – the only difference being that mail would be delivered to a mailbox outside of the Exchange Organization.
Similarly, a mail user in Office 365 / Exchange Online would allow a user to logon to access various hosted resources (albeit not a mailbox, as the mailbox is outside of the tenant). An administrator would be able to grant access to all of the various resources in Office365 (Teams, Sharepoint, Office licenses, etc.) all while maintaining the user’s mailbox outside of the tenant. In any event, there is much more in Office 365 that can be accessed than just a mailbox. If you were to attempt to logon to any of the other services in Office 365 with that mail user (assuming they were sufficiently provisioned), I believe you would better understand the purpose.
For more information (though outside of the first paragraph, it mainly pertains to how to create and manage them), please review this article; relevant information follows:
Mail users are similar to mail contacts. Both have external email addresses and both contain information about people outside your Exchange or Exchange Online organization that can be displayed in the shared address book and other address lists. However, unlike a mail contact, a mail user has logon credentials in your Exchange or Office 365 organization and can access resources. For more information, see Recipients.