When you run the first DirSync synchronization, Office365 creates the mailboxes in Office365 and assigns a primary SMTP address of user@lucernepublishing.com. It will also create a secondary address of user@lucernepublishing.onmicrosoft.com. Users can now log on to Office365 and access their mailboxes.
If you then set up either password synchronization with Dirsync or implement SSO, typically using Active Directory Federation Services (AD FS), then users can log on to Office365 using the same credentials that they use for on-premises logins. In the case of password sync, there are still two separate accounts, one online and one in the cloud, but they have the same user name (user@lucernepublishing.com) and the password is synchronized between the two environments.