UPDATE: this problem is fixed in the Windows 10 Fall Update (version 1511, build 10586)

A bug in the Edge Browser used by default in Windows 10 disables the Drag & Drop to upload feature, both when logged in and when using Customer Upload.

Until Microsoft fixes the bug workarounds include using the browse button and selecting files from a list, or you can open the same page with Internet Explorer just by clicking or tapping the More actions icon in Edge (the ellipses at the top right) and then select Open with Internet Explorer.

Non-Microsoft browsers such as Chrome or Firefox used in Windows 10 do not experience this problem.