Some of the changes made in Easy365Manager are made directly in Office 365 (e.g. mailbox quotas). Other changes are made to the user attributes in your on-prem AD (e.g. mail alias).
Using Easy365Manager you have the option to immediately trigger a synchronization to Office 365 after making changes.
Simply tick the “Synchronize AD with Office 365” check box in the bottom left corner of the Office 365 tab before clicking OK or Apply: