In the menu bar at the top, select View. Scroll down in that View menu and select Developer. And then in the little pop-out sub-menu, select Developer Tools.
Tip: The Developer Tools window is associated with the current Brave Browser window, and NOT associated with other Brave Browser windows.
Also, the first occasion of using Developer Tools, the Developer Tools “window” is usually a part of its associated Brave Browser window . . . but you can make the Developer Tools window, a standalone window (though keeping its association with the Brave Browser window). I recommend that you do so:
In the upper right-hand corner of the Developer Tools window, there is an “Organizer” 3-vertical-dots button - click on that. The result should be a pop-up:
Notice the tiny blue icon that is first (left to right) among 4. Click on that icon. The Developer Tools window will become a standalone window . . .
THAT is easy to manage.
Enabling “Developer Mode” does NOT open the Developer Tools window.
Enabling “Developer Mode” allows you to monitor - live - via the Developer Tools window’s Console tab, browser activity in addition to all “console.log” reports from loaded-and-enabled extensions.