Before the advent of modern monitors, CRT technology was used, which was vulnerable to screen burn-ins. Screensavers avoided these problems by producing animations instead of persistent images.
Were constantly changing pixels more effective than a simple black screen to avoid these burn-ins, or were there just for entertainment?
To elaborate on Jon Custer's comment: yes, for protecting the display against burn-in, a black screen would do just as well or better as any "screensaver". However, the graphical effects displayed by screensavers had two important purposes:
They showed that the computer was turned on and functional.
They looked nice.
Purpose #1 one used to be a lot more important than one might assume from a modern perspective, in large part because up until the mid-to-late 90s most computers had a mechanical power switch with no software protection.
Thus, if you walked up to a computer that you assumed was turned off (because the screen was dark) and pressed the power button to turn it on, you would instead turn the computer off immediately, potentially losing any unsaved work.
There would be no popup asking you whether you really wanted to turn off power, and indeed typically no way for the OS to even perform a controlled shutdown. The button would just turn off the PSU, leaving the CPU and the disk drives etc. with no power, and down the system went.
In the worst case, turning the computer off at the wrong moment could even lead to data corruption, if you happened to interrupt some critical operation such as disk defragmentation — which, coincidentally, was also a fairly common long-running process that you might well start up and then leave running while you went to grab a coffee, or maybe even overnight.
Even when the ability for the OS to intercept the power button press was added (precisely because accidental power-off was such a common and potentially destructive mistake), holding the button down for a second or two would typically override it and force a hard power-off (because there was still a need for a way to be able to turn off and reboot the computer even if it got completely frozen and unresponsive, which was unfortunately all too common). Coincidentally, if the screen was completely turned off and in power save mode, it might take a second or two to turn back on…
Having a graphical screensaver running was an easy way to prevent these kinds of mistakes, simply by immediately and intuitively showing anyone who intended to use the computer that it was already running, and that they should keep their fingers off the power button.
Of course, purpose #2 should also not be neglected. Screensavers were simply cute, and often showcased the graphics capabilities of the computer (such as they were, at least). Many people would download and install fancy custom screensavers just to personalize their computer or to show off what it could do.
(Indeed, some of those "screensavers" wouldn't even do anything to protect the screen from burn-in, and could even contribute to it by displaying static background images or e.g. a repeating slideshow with no change to the image positions on the screen. But often it wasn't a big problem anyway — as time went on and display technology advanced, actual burn-in became less and less of a risk.)
Computer and OS manufacturers would also use fancy screensavers to showcase their products, particularly since computer stores would often have demo computers running for customers to try out, and naturally those computers would spend a lot of their time idle and showing a screensaver. I'm sure the Flurry screensaver on OSX, for example, got Apple a bunch of sales in the early 2000s just by looking a lot fancier on the store shelf than anything likely to come preinstalled on a Windows box at the time.
A blank screen would have been just as effective for protecting the CRT from the burn-in associated with a static display. However, for a time in the late 80's through 1990's, certain popular screensavers would become highly-recognizable, iconic, imagery cemented in the brains of the by-then-mainstream-computer users.
More niche computers, like the Commodore Amiga and Atari ST of the time, were known for their demo scene productions, which were often an out-growth of widespread software cracking. For more mainstream users with Windows PC's or Apple Macintosh, these demos were nearly non-existent. So, there was an unfulfilled "need" on mainstream computers for applications that had no point beyond showing off the machine's graphical abilities. Screen savers filled this need nicely, while also inventing a raison d'etre and marketing angle about protecting your expensive CRT. Later, when people also became more aware of computer security, a second purpose of locking-out unauthorized users also materialized.
The commercial After Dark software was a very popular mainstream computer application in this period, with the "Flying Toasters" screensaver probably being the most iconic. Another example was the selection of screensavers included with Microsoft Windows, which could be found running on nearly all office computers for many years in the 1990s.
So, there was a reason for more interesting, graphical, screensavers, even though it was predominately for entertainment and marketing. It's always been the case that computer generated "art", such as demo scene productions and screensavers, have an aesthetic value, and can exist and even become wildly popular just because...
A common usage pattern in the 1990s was that someone who was stepping out of the office but would either stay nearby or else be back soon would trigger the screen saver, but someone who was leaving for an extended period of time would shut off the monitor. Thus, a quick glance at the screen would indicate whether one should look nearby for the person in question.
Were constantly changing pixels more effective than a simple black screen to avoid these burn-ins,
No. in fact, every use will degrade the coating, thus a straight black screen would be better han doing whatever animation
or were there just for entertainment?
Exactly. And like with all entertainment, as soon as people are willing to pay for it, more and more offers will pop up.
Note that at the time, most monitors did not have any sort of power saving modes. The system could blank the signal to the monitor, but the power used was approximately the same. The ability for the CPU to reduce power was also very limited.
That makes the marginal cost of the animations quite small and a black screen very dull. Unix X windows had a basic "screenblank", but we would always select one of the other modes because it looked better. (In my case, it wasn't necessary to have it visible to show the system was on, it was just because it seemed more interesting).
Once you get a sizable population of monitors that can power down and systems that can trigger the power savings, the utility of the screensaver displays becomes much lower.
I've been told that the filament wears faster when starting from cold state during warm up. It was therefore preferable to not switch off, switch on the tube all the time (and I remember in my childhood when valve TVs were still state of the art, that startup was indeed a big deal and that the TV technician firmly advised to not power cycle too often). The most acute issue was phosphor burn in so just having a changing picture to avoid it was more important. From there, it became in and itself a software category for novelty and entertainment.
