When a user follows a site, a link to that site is added to the user's Followed Sites list. If you're using both SharePoint Server and SharePoint Online, your users will have different followed lists for sites in each location. With SharePoint hybrid, you can consolidate the information from both locations into the SharePoint Online list in Office 365.
How hybrid site following works
When you enable hybrid site following:
The Sites link in a SharePoint Server site is redirected to Office 365 (SharePoint Server 2013).
The Sites tile in the app launcher is redirected to Office 365 (SharePoint Server 2016).
When a user follows a site in SharePoint Server, it is added to the followed list in both SharePoint Server and Office 365.
While the SharePoint Server followed list continues to be updated, users are directed to the followed list in Office 365, which contains followed sites from both locations.
The SharePoint newsfeed functionality is unaffected. Users will continue to have separate newsfeeds in SharePoint Server and Office 365, and each will show activities for sites and documents for SharePoint Server and Office 365, respectively. Also, follow documents functionality remains unaffected, and follow people functionality remains in SharePoint Server only.
Note that existing followed sites lists in SharePoint Server are not migrated to Office 365 when you turn this feature on (though any sites in the Office 365 list will remain there). Users will have to follow their SharePoint Server sites again once the feature is turned on.
Setting up hybrid site following
Hybrid site following is part of Hybrid Sites Features, and is available on both SharePoint Server 2013 and SharePoint Server 2016. See Hybrid sites features and OneDrive for Business for information about the other features included with Hybrid Sites Features.