How can I use "Azure File Storage" with Web App Service?

Yes, you can. In Linux Web Apps and Windows Containers Web Apps only.

In the Web App, under Application Settings you will find Mount storage (Preview) section where you can mount Storage account. Works like a charm.

If you're looking for mapping a drive to your Azure File Service share in your Web App using SMB protocol, then I don't think you can do as of today.

What you could do is make use of Azure File Service REST API and manipulate shares and files that way. You don't have to actually use the REST API per se; you can simply use the latest version of .Net Storage Client library which is a wrapper over REST API.

Please see this link for more details on how to use this using Storage Client library: https://azure.microsoft.com/en-in/documentation/articles/storage-dotnet-how-to-use-files/.