I hooked up a Mirabox HDMI capture card with loop out between my Xbox 360 and display. It passes the feed through to my MacBook Pro. Both the video and audio pass through cleanly to OBS and I've been streaming successfully to Twitch with that setup.
I then wanted to add my headphones and Blue Yeti microphone into the setup so that I could hear the game audio and talk during the stream.
Despite OBS picking up the audio through the Audio Input Device, I couldn't hear it in my headphones. That's because by default Audio Monitoring is turned off for audio devices.
If I want to be able to hear the sound and pass it through as output to OBS, I have to change the Audio Monitoring setting for that device. In the Audio Mixer section, there is a gear icon next to the audio device. I click that and choose Advanced Audio Properties. From there, I change that device's Audio Monitoring setting from Monitor Off to Monitor and Output.
Now I am able to hear the audio that is coming through Mirabox from my Xbox. And anyone watching my stream can hear it too.