This is where it all comes together. At last, from: https://azure.microsoft.com/en-us/blog/score-one-for-the-it-pro-azure-file-sync-is-now-generally-available/
Azure File Sync replicates files from your on-premises Windows Server to an Azure file share, just like how you might have used DFS-R to replicate data between Windows Servers. Once you have a copy of your data in Azure, you can enable cloud tiering—the real magic of Azure File Sync—to store only the hottest and most recently accessed data on-premises.
And since the cloud has a full copy of your data, you can connect as many servers to your Azure file share as you want, allowing you to establish quick caches of your data wherever your users happen to be. As mentioned above, in simple terms, Azure File Sync enables you to centralize your file services in Azure while maintaining local access to your data.