I just opened a free Azure trial account and went through the various forms to create a test web app. I would like to edit the source files locally with Visual Studio 2015 howev
You can download your project file using KUDU site: https://blogs.msdn.microsoft.com/benjaminperkins/2014/03/24/using-kudu-with-windows-azure-web-sites/
Select Debug console -> CMD from there you can go inside site -> wwwroot and select download button to the left of any folder:
It will zip up all the files under that folder and start downloading.