Upon connecting a Blackmagic encoder to your Mac, you will see it show up in Ecamm Live. However it may not be selectable in the menu. (It will be gray.) This means that Ecamm Live has not received any video from the device. To troubleshoot this:
Ensure that you have installed the latest Blackmagic Desktop software. You can download the latest Blackmagic Desktop for Mac here on their website. Click the Mac OS X button in the Blackmagic Video section in the left-most column of the page. A registration form will show up. You do not need to fill it out. (Look for a tiny Download Only. button in the bottom corner.)
For macOS 11 Big Sur, you may need Blackmagic Desktop Video 12 software package. Visit their downloads page here and look for Desktop Video 12.0 or newer in the left-most column.
In Blackmagic's Desktop Video Setup app, (found in your Applications folder), ensure that you've picked HDMI rather than SDI in the device's settings. This is pretty much the only setting you'll ever need to change, and once changed, it should stay set to HDMI.
If your are using an Thunderbolt 3 to Thunderbolt 2 adaptor, ensure that you're using Apple's adapter for this. Some multi-port adapters look like they have a Thunderbolt 2 port, but unfortunately they only work for connecting a MiniDP display, and will not work for Thunderbolt 2 devices.
Ensure that your camera is outputting an HDMI resolution and frame rate supported by the Blackmagic encoder. (It may not support some higher frame sizes and frame rates.) See the details here for Mini Recorder. For example, it does not support 1080p60 or anything above 1080.
Quit Ecamm Live and check to see if it's working in the Capture tab of the Blackmagic Media Express app. (Typically this can be found in your Applications folder after installing the Blackmagic Desktop software.) Ensure that you quit Ecamm Live before running Blackmagic Media Express and vice-versa, as only one app can access the Blackmagic device at a time.
IMPORTANT: If you are not able to see a feed from the camera using the Blackmagic Media Express app (step 6 above), this indicates that the Blackmagic device is not working, and we recommend working with Blackmagic's support team before contacting Ecamm.