How to prevent my screens from turning off, when the screen is locked?

Open your System Settings. We need to make changes to Power and Brightness & Lock settings.

Power Settings

Change the value of Suspend when inactive for to Don't suspend.

Brightness & Lock Settings

Change the value of Turn screen off when inactive for to Never.

This should help to achieve your desired result.

I had to install additional screensaver program XScreenSaver and use it instead of default one because accepted answer had no effect for me on Ubuntu 16.04 and 17.10.

1. Install XScreenSaver:

   sudo apt install xscreensaver
   

2. Add this command to Startup Applications:

   xscreensaver -nosplash
   

3. Disable default Super+L shortcut

   • Install CompizConfig Settings Manager

     sudo apt install compizconfig-settings-manager
     

   • Launch it and click "Ubuntu Unity Plugin"

   • Click the button next to "Key to lock the screen."
   • Disable

4. Add this command to enable Super+L shortcut for XScreenSaver in System Settings > Keyboard > Shortcuts > Custom Shortcuts

   xscreensaver-command -lock