I've seen that a few other people have had this same problem, so I followed the instructions posted, but am still having the same issue.
I downloaded Evernote from the web, and installed it that way.
I downloaded Evernote Web Clipper from the web (called Evernote.5.9.10.safariextz)
I move the installer onto my desktop.
Then, I double-click the installer, and it says "Evernote.5.9.10.safariextz is an appliation downloaded from the Internet. Are you sure you want to open it?"
I click "Open."
Then I get a window that says "Are you sure you want to install the extension “Evernote Web Clipper”?" and more.
I click "Install."
Then the icon disappears, and nothing happens.
I got rid of my old Evernote, just in case I had originally installed it from the App Store.
I deleted the files from the trash.
I reinstalled from the Web page, and tried again.