Hey there, Silver10,
Thanks so much for using Apple Support Communities. We want to help get your MacBook working as expected with your external display.
To start, we want to recommend that you pull this article up: Use your TV as a display with Mac - Apple Support. That resource is going to have the complete process of connecting your MacBook to a TV for use as an external display, and should do the trick for you here. If you've followed those steps and you're still not able to see your Mac desktop on that screen, then you'd want to follow the steps here to try and resolve that:
1. Close the lid on your Mac laptop to put it to sleep.
2. Connect the power adapter to your Mac and a power outlet.
3. Reconnect your Mac to an external display.
4. Connect a USB keyboard, mouse, or trackpad to your Mac.
5. If you’re using a wireless keyboard, mouse, or trackpad, make sure it’s turned on.
6. Wait a few seconds, then press any key on the external keyboard, or use the mouse or trackpad, to wake your Mac.
7. If your wireless keyboard, mouse, or trackpad isn’t connected (paired) with your Mac, Bluetooth Setup Assistant opens so you can connect them.
That info can also be found here: If you can’t see your Mac desktop on an external display - Apple Support
We hope that this information is helpful.
Have a great day!