Whether your site is using Forms authentication or Windows authentication without impersonation should not matter - the same account (Network Service by default) is used to access resources from the site. If this account has read/write permissions on the physical application files, the file explorer should work.
If you are using windows authentication with impersonation, then the application users will also need to have read/write permissions on the folders, because the server will use their identity to perform operations and access resources.
Finally, your application must be running under Medium trust or higher. Lower trust levels do not allow reading/writing files in the application folder. If you must run your application in a low trust level, then we suggest that you add the Telerik.Web.UI assembly to the server's GAC. This way it will run in Full trust, while the rest of the application will run under your trust level. This will allow the FileExplorer control to work properly.
the Telerik team