The cathode ray oscilloscope works by using an electron gun to shoot a beam of electrons at a phosphorescent screen. The beam is controlled by deflection plates that bend the beam horizontally and vertically based on input signals, causing the beam to trace out waveforms on the screen. Key components include the electron gun, deflection plates, time base circuitry to control the horizontal sweep, and a trigger circuit. Calculations can be performed on waveforms by relating the timebase and voltage divisions to measurements taken from the traced signals.