Google Places API - Free Requests per day after enabling billing?

Places now operates using a different pricing scheme, depending on which SKU is used during a user session, see here. For the original answer to the asked question, which was valid before June 11, 2018, see below.

From reading the link you provided, I think you may have misinterpreted the billing scheme for the Google Places API. The free default tier initially behaves this way:

The Google Places API Web Service enforces a default limit of 1,000 free requests per 24 hour period

After this:

You can increase this limit free of charge, up to 150,000 requests per 24 hour period, by enabling billing on the Google API Console to verify your identity.

So in other words, you can increase the limit to 150K requests per day, completely free of charge. To do this, you need to enable billing:

Verify your identity to get 150,000 free requests per 24 hour period, by enabling billing on the Google API Console. A credit card is required for verification. We ask for your credit card purely to validate your identity. Your card will not be charged for use of the Google Places API Web Service.

From what I read, you will not be charged so long as you do not exceed the 150K request daily threshold. After that point, you would be charged.

New policy of Google Places API

"Requests per Day (QPD) limits have ended, effective June 11, 2018"

Reminder: To use our Maps, Routes, and Places APIs:

You must enable billing on each of your projects.