In the Google Developers console, if I choose my API project there is a Quotas tab. Under Billing Status there is a \'Apply for a higher quota\' link which links to a page that
Sounds like you're talking about this: https://developers.google.com/places/web-service/usage#verify_your_identity_by_enabling_billing
Once done, it'll ask you for your credit card information. Once that's all submitted, what I did was to set a budget of $0 per day to be sure they never billed me anything.
The $0 budget is likely unnecessary since they say they'll won't bill you in their documentation unless you upgrade to premium... but I like that extra level of surety.
After adding enabling billing, hit the top-left menu item and click on "API Manager". You should see "Google Places API Web Service" listed. Click on that, and it should tell you your quota of how many requests you've used out of 150,000 possible.