There is no need to download a separate application. You can do the same thing by copying the text into the Notepad application on the Windows operating system or the TextEdit application on the Mac. In TextEdit, be sure to click Format > Make Plain Text.
Alternatively, after you paste into Word, highlight the text in question. You can then click [Ctrl] + [6] on Windows or [Command] + [6] on the Mac to strip out any field codes. This will remove the link between the citations and EndNote that was responsible for the field shading (grey background) you are seeing.