This is dependent on the way you set up the browser. Some browsers automatically open and correctly associate the enw file, and some don’t. Some download it and you need to open it.
You might want to download (from the right hand bar of this KB article linked below) the the Little Endnote How to Book, which is full of otherwise obscure information about endnote. Being a old dog, I really miss manuals!
http://endnote.com/kb/129310 (fixed link)