Back in the time of monochromatic monitors ( you can also read Black and White , but some times these monitors are also green instead of white ) when you leave the computer on for a long time with same text on the screen the phosphor on the screen will be continuously exposed to the same high intensity beam of electrons from the gun.
So there is a risk of damage to phosphor that will show up as a burned image of the characters on the screen even if the screen is turned off.
For programs like wordstar there was a permanent menu shown on the top of the screen that added up to the burning process. Even if you turn off the monitor , you can read most of the commands of wordstar .
To cop up with such situation , TSR ( terminate and Stay resident Program ) were written for DOS environment. They were called screen savers. As evident from the name , their duty is to protect the screen from burning by creating a moving pattern when you are not working on the machine.
When color monitors were introduced to the market , the screen savers was no longer needed because the image you are viewing on the monitor is not a single image but the effect of continuous scanning of three guns of primary colors R , G and B . So even if you are seeing a constant image , the screen is being continuously changed. Also the quality of the phosphorus coating is also increased to offer more tolerances to the burning process.
And with the advent of LCD monitors , the basic way of image display have been changed and we no longer need the screen savers .
So you don’t need screen savers anymore to save your screens
But from the earlier time of a moving dot on the screen , the screen saver has grown to a big industry, using your computer screen as a bill board for the new products and using your spare processing time when you are not using your computer .For example the projects of distributed computing process like SETI . So if you are downloading a free screen saver from internet and installing it on your computer , most probably you are agreeing to be part of such projects who use your spare computer power for their projects for good or bad intentions .