on this page (https://developers.google.com/youtube/v3/getting-started) google says:
You can find the quota available to your application in the Developers Console.
I went to my Developer Console, and it shows me how many requests I made, but it does not show my exact quota (for my free account). Does anybody know how to access this information? The url for the developer console is https://console.developers.google.com/
Thanks!
This will give you a generalized view of your usage over time for that particular API.
Update for July 2020
Check out the IAM Admin/Quotas page for a simplified view of your usage against your total quotas.
You can change your project in the left of the top bar and search per API with the dropdown options on the page.
Clicking any of the listed quotas will show you additional metrics, e.g. a graph of requests over time.