The JavaScript Debugger is sort of a re-launch of a ServiceNow feature that was removed in the Fuji release. Since then, the feature has undergone a major overhaul before being reintroduced, resulting in the functionality that we now have access to in Istanbul.
The Debugger can be opened by clicking on the JavaScript Debugger button in the list of icons above any Script field, such as those above the Script field on a Business Rule.
Nearly everything relating to debugging happens within a specific session, and the JavaScript Debugger is no different. In order for the debugger to interact with a script's execution, it must be open in a browser window that's within the same interactive session (aka, user session) as the/ session in which the script is being executed on the server.
Of course, code execution will also only be debugged if at least one breakpoint is set, and if the JavaScript debugger is indeed open, and switched on. The debugger launches by default, in an On state. Clicking the small pause icon at the top-right of the debugger will turn it off, at which point you can click the I/O icon to turn it back on.