Sorry.
(Someone should do a better version.)
A CRT screen basically works by firing electrons (or if you want that a bit more scaremongery: a beam of beta radiation) at the phosphor plate, the impact of which makes it emit light. Shades are achieved by varying the intensity of the electron beam and colour is formed by having three different colours of phosphor dot and three beams (and a grid to make sure the right dots were hit by the right electrons), but these details are irrelevant to the burn-in problem.
See https://en.wikipedia.org/wiki/Cathode-ray_tube for a longer and better explanation of how it works, and the history of the technology.
Under normal operation this is mostly all fine: the phosphor "recovers" once the beam stops hitting it for a while. The phosphor does degrade (essentially they burn) a little over time with use which is also fine if the wear is level, though it does mean the screen effectively fades a little with age. If you leave a particular image on screen for a long time then it becomes more of a problem because the lit areas degrade and the unlit ones don't. Essentially the phosphor plate develops permanent tan lines. It was not uncommon to see the shadow of login prompts or semi-permanent UI elements burned into screens this way.
It happens to other screen types too, though for slightly different reasons in each case. Plasma screens are particularly sensitive to the problem, but it can affect LCD displays too (some types worse than others). See https://en.wikipedia.org/wiki/Screen_burn-in for examples and a bit more detail.
The earliest screen-savers were just turned the screen black. This could lead to machine being powered off accidentally (potentially losing work if a user had walked off without saving) when someone thinks they are off and tries to turn them on, so instead a simple moving message was added, the message to indicate the machine is intentionally on and the movement to stop the message burning in. Other animations were added, with After Dark's iconic flying toasters appearing in ~89, and around the Windows 3.x era it exploded and almost became an art form (anyone else fondly remember the three-and-two-tenths flying toaster brigade, or Johnny Cast-Away?), some people competed to have the best screen-saver, or just the most "them" screensaver along with wallpaper/colour-scheme/other-desktop-customisations, and of course every corporate desktop carried the (bouncing) company branding when idle. Things died down when we became more energy conscious: going back to black screens because turning the whole apparatus off when idle saves energy.