1. Download BlackHole tool on existential.audio/blackhole. This tool is open source, decide yourself if this is safe for you (we do not guarantee anything). We downloaded the 2 channel version. Install BlackHole on your Mac (this creates an audio input and output channel with your system audio).
2. Open the app Audio MIDI setup. Click on "+". Add an Aggregate device. Add BlackHole and your microphone. Rename the Aggregate device to "Mic and BlackHole" (now you have an audio channel wich integrates two audio sources: what goes into Black Hole and mic audio; this will make it possible to record system audio in your screen recording).
3. Open the app Audio MIDI setup. Click on "+". Add a Multiple output device. Add BlackHole + your speakers (or headset). Rename the Multiple output device to "Speaker and BlackHole" (now you have a channel to redirect your system audio to Black Hole and your speakers).
4. Go to System Preferences / Sound / Output, choose "Speaker and BlackHole" as sound output device (this is necessary to finally redirect system audio to Black Hole and your speakers).
5. Open Quick Time Player, go to File / New screen recording
6. Go to Options, choose "Mic and BlackHole" as Microphone
7. Start your screen recording!
8. Optional: To better the input sound mix between your mic and Black Hole you can change the input volume in the Audio MIDI setup app. Just choose Black Hole or Mic and change the Master stream volume.

Note: I couldn't change the volume on my Mac-Speakers after choosing "Mic and BlackHole" as Microphone (step 6). If you need to change the volume you will have to do this in MIDI-setup. I couldn't find another way to solve this.


How it works (illustration):

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Idea on how to do comes from: