Microsoft® Exchange relies on Active Directory to keep track of servers, users, and all Exchange objects. Active Directory uses the term "domain", but this is not the same as a domain as referenced by DNS (such as example.com). For Exchange, each server and each mailbox is a member of an Active Directory domain (and in our environment, these Active Directory domains are shared by multiple customers).
The reason the Exchange Domain is important is that when you attempt to log in to your mailbox, or SharePoint, or access any other hosted Exchange item, you need to verify your identity and authenticate against this domain. This is not specific to Smarsh Hosted Services, but is the architecture used by MS Exchange.
If you need to find your Exchange Domain, you can log in to CONTROL PANEL and navigate to Get Started Here! > Advanced Settings.