The steps for setting up the API key in the Google developer console are tricky. Be sure to:
1. Create a *browser* key
The steps are described here:
Try your key first with the referrer field left blank. Note that it can take up to five minutes for changes to propagate from Google.
If you are testing on a local server (localhost) instead of a public server, leave the referrer blank permanently, otherwise you can fill it in once the maps are working.
If the above don’t help, try press F12 to see a more detailed error message in your browser. If you can provide a URL with a map in your post, I can take a look as well.