EDIT - I see now the jeffreythefrog beat me to it - and with detailed pictures, too. Just follow his advice.
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First of all, you CAN record your screen with audio.
Second: to be more specific, are you trying to record audio that is being played on your mac? This requires a little work, but it can be done. If all you want is record other audio captured by the microphone, that is easier.
I am assuming that you mean the former.
This will require a piece of additional software, but then you will record using the same Quicktime Player "New Screen Recording" or directly with Command-Shift-5 in the Screenshot application.
1) Download and install BlackHole Audio Driver
2) This adds some new audio input and output options, which you will be able to see in the menu bar or in System Preferences->Sound
3) You can now use these two "BlackHole 2ch" virtual devices to route audio output from one application to the input of another
4) You can set the system output to "BlackHole 2ch", and in Quicktime Player select the audio input as BlackHole 2ch, and record
BUT doing it this way, you won't be able to hear anything, as you record.
To be able to monitor the audio via the speakers or headphones at the same time it's being recorded, you need to be able to send the output to two devices at the same time.
This requires creating a Multi-Output device, in Audio MIDI Setup (this is a program already on your mac, in /Applications/Utilities).
5) Open Audio MIDI Setup, click the + button at the bottom left and choose "New Multi-Output Device". Add both "BlackHole 2ch" and the speakers (or headphones) to this new device.
6) Proceed as in 4), but now for system output you should select the new Multi-Output Device created in the previous step.
You can now monitor and record the audio at the same time.