The browser method is not a ServiceNow-specific way of debugging client-side scripts, but can be a useful one. Because these scripts are running on the client, web browsers often have ways of displaying running JavaScript code that you can check.
These methods vary based on the browser you use to view ServiceNow, but most modern browsers will have some way of checking what JavaScript is running which can be used to test your client-side scripts.
I personally tend to like using Google Chrome for browser-based testing, but other browsers can be just as good.
