You add that code in the </head> section of a theme component. You can read more about that here.
Discourse is a single page application. So, the implementation might differ from the documentation that Statcounter provided. Your implementation has to account for that; otherwise, you’ll only get the stats for the initial page view.
Here’s an example of how we send stats on page transitions for Google analytics (this is built-in, and it only starts working as soon as you add your Google analytics credentials in the admin settings).
You’ll also need to make sure that the scripts from Statcounter are (CSP) allowed either via your theme component or in your site settings.