The built-in way
Recording with the ChromeOS screen recorder
Press Shift+Ctrl+Show windows, the key with a rectangle and two lines, or open Quick Settings and choose Screen capture. Switch the toolbar from photo to video, pick full screen, a window or an area, then open the settings gear and turn on the microphone. Recent ChromeOS versions also offer to record the device's audio from the same menu.
Press record. The file is saved to your Downloads folder as WebM with your system cursor in it, and there's no editor, so trims, zooms and volume changes need another app.