Clicking it advised me that, although I had the latest Java version for Mac (v. Java 8 Update 51), it was being blocked on my browser (Safari). I deactivated the "AdBlock" extension on Safari, closed and reopended Safari, and opened Wordle on Safari. I entered my test text entry in the Wordle text container, pressed Go, and was asked if I wanted Java to work on the Wordle site. I clicked "OK" and Wordle finally activated nicely.
I hope that helps others have difficulty activating Wordle on Java-enabled browsers. (I wish the troubleshooting process was more intuitive and easier to follow so we could get Java to work with the least of effort.)
Great app, Jonathan! Thank you very much.