Treat such a request as the world's most misleading "permission denied" error.
When WordPress fails to write to a file due to a permission error, it has some very old code that tries to work around the problem by asking for FTP information. But these days FTP is dangerously insecure and unsupported. Do not provide your NearlyFreeSpeech.NET credentials; it won't work and could potentially get you hacked.
If it happens during a media upload, review the "Enable file uploading" section of our Installing WordPress guide. It will show you how to set your site up with the permissions to make that work.
This also comes up if you attempt to have WordPress update itself over the web. Don't. It requires major security compromises that will turn the next vulnerability discovered in WordPress, PHP, your blog's theme or any active plugin into a complete takeover of your entire site. We will not assist you to do that.
To safely update WordPress and its themes and plugins, use the wp-cli command line tool, as documented in the "Upgrade WordPress" section of the Installing WordPress guide.