To begin the Discord stream, click Start Virtual Camera under Controls in OBS. Finally, return to the Discord app and connect to a voice channel (create a private voice channel to test it out first).

But, like me, if you only want to use OBS to reframe your webcam, select the I will only be using the virtual camera option. It's just quicker to start using the app this way.

Then, whichever app you want to use a webcam with, instead of selecting your hardware webcam, choose the OBS Virtual Camera option instead.