What was the original purpose of screensavers?
Screen savers were initially introduced to prevent a phenomenon known as "phosphor burn-in," which permanently damaged cathode ray tubes.
Before LCD monitors were introduced, cathode ray televisions were used as display units for computers. A cathode ray tube has a system that emits electrons on a screen which is coated with phosphors and these emit light whenever they are struck by an electron that has a sufficient amount of energy. Color cathode ray tubes produce the effect by using phosphors that emit light of different colors.
Early models of cathode ray tubes had a problem that resulted in the phosphors getting damaged if the same image was displayed for a long duration of time. When these were used as computer display units, the same image was displayed when the computer was not being used. To avoid the screen from getting damaged when this happened, screen savers were used to display a moving image. This prevented the screen from getting damaged or phosphor burn-in taking place.
The answer posted above is correct and clear. However it is only a partial answer to the question.
One of the ways of protecting the video display unit (VDU) of computers from phosphor burn-in is to switch-off the VDU. But this simple method has a disadvantage. A Cathode ray tube (CRT) based VDU requires some warm up time after the VDU is switched on to start displaying the image. Switching off a VDU when there is no need to view it results in cooling down of the CRT, so that some additional time after re-switching is required before the VDU start displaying the images again.
Screen saver is a device or method that combines the requirement of protecting the screen from phosphor burn-in, and of keeping the CRT hot enough to start re-functioning instantly when required. This is achieve by an image that changes and moves over the screen in a systematic way so that at a time only a small part of the VDU is lit up with an image at a time, and that over a complete cycle of the screen-saver image movement, the entire screen receives uniform intensity of energy. This ensures that the CRT is not cooled down to a level that is below the requirement for immediate start. Also the phosphor burn-in is considerably reduced because the total energy impacted on any one part of the screen is reduced considerably.