When people talk about a hybrid cloud storage solution, it typically means they want to keep using the existing file server. At the same time, they also want to replicate the file server data into cloud storage and make it available to remote users.
Basically, the local employees will keep using the local file server. The remote users will be using the cloud storage. The local file server and cloud storage will be in sync. If the file is changed locally, it is uploaded also to the cloud storage so remote user can use. If remote user changes the file, the change will also propagate back to the local file server.
It serves many purposes. First the local file server and cloud storage are like a backup system, backing up each other. Second, it is a VPN-replacement system, enabling remote users to access files without VPN. Third, it is an access system, it enables mobile users with iOS/Android and Windows Phone devices to access the files and folders.