Hi,
probably you need to enable billing in the Project.
"probably" because to be sure about the problem I need to see your site, so if you have already enabled billing then please post me your site link to check this.

Google Maps is free?
Google maps should give to all customer 200$ of free credit for each month, so it is free for 99,9% of customers. But to works it require that you enable billing and associate a payment method with your account.

Hi,
error is:

This API project is not authorized to use this API. Please ensure this API is activated in the Google Developers Console: https://console.developers.google.com/apis/api/places_backend?project=_ For more information on authentication and Google Maps JavaScript API services please see: https://developers.google.com/maps/documentation/javascript/get-api-key

Have you enabled APIs described at http://docs.easy-profile.com/index.php/article/google-maps-api-key (Step 2)?
if you have followed video tutorial then have you also enabled APIs in the message? (see screenshot)

the problem persists, and we cannot get it to work.. what are we doing wrong?

Hi,
we see still same error, have you enabled following APIs?
- Maps JavaScript API
- Places API
- Geocoding API

you can see it from Project dashboard (see screenshot)