Found it! I had to activate (not only generate) the API key under https://console.developers.google.com first, then it worked out.
But again: Is the API key mandatory?
I double-checked ist, my API key has no spaces before or after.
Is the API mandatory to get the plugin started? I followed all steps in the Vimeo video and I understood adding the API key is optional.
https://vimeo.com/77878709
But anyway, it doesn’t work also with the API entered.
I have exactly the same problem here.
No Google Fonts are shown, only the – Theme Default – Standard Web Fonts.
My environment: WordPress 4.3, Theme: Bloggo