Some part of the enjoyment of a movie, book, or video game is remaining in suspense and feeling the shock when a secret is finally explained. But in any narrative worth wading through, this is only a small part, hence the appeal of plowing through them over and over if they're that good. Therefore, the panic about revealing spoilers means that this is the only possible source of enjoyment -- don't rob us of what little potential is there!
Now, enjoyment is a function of both the product and the consumer. Maybe spoiler-phobia means the product is otherwise garbage and its sole appeal is a shocking reveal. However, it could also mean the product is pretty good, but the mindset and personality of the consumer is so incapable of appreciating any aspect other than the acquisition of new information, as though following narratives were like flipping through someone else's PowerPoint slideshow for the gist and the all-important "take-home message."
Steve Sailer guessed that spoiler-phobia began in full force with the release of The Sixth Sense, and there was indeed a real hysteria about that one. Here's one example from an NYT review:
At first, the doctor doesn't believe the boy. But then, well, let's not take the story any further lest its colossally sentimental payoff be compromised.
The first appearance in the NYT of the term "spoilers" in this sense actually came earlier that year, referring to newly added scenes to the original Star Wars movies when they were re-released in theaters. Dictionary.com has an entry from some jargon databank that shows it appearing on Usenet in 1995.
So, like all other infuriating aspects of today's culture, spoiler-phobia began just after the violence level began plummeting in the early-mid 1990s, causing everyone to focus on more trivial matters, throwing overboard their earlier appreciation for the grand and the sublime. Again, that could be due to the movies themselves becoming more empty, the audience members becoming more airheaded, or both (yep).
Even though the exact term "spoilers" doesn't appear earlier, perhaps the same panic appeared but using different words, as with the Sixth Sense review above. Note: not just a lack of spoilers, but a conscious paranoia about even hinting at spoilers, and a clear declaration that there will be no spoilers in the review. I tried looking for NYT articles on The Empire Strikes Back and Psycho from the year they came out, but they're all behind a paywall. I could do a Lexis-Nexis search, but I don't care that much about this topic. Someone else can go for it.
I have little personal experience to relate from the movie culture of the 1980s and before, as I either wasn't born or watched more movies on home video than in the theater (though I saw lots there too). I don't recall any such paranoia, though. There was one popular video game, however, that had a shocking plot reveal at the end -- that the bounty hunter who's been kicking alien butt is really a woman under the suit of armor -- and yet there was no hysteria about it. In fact, one of the first password codes that we learned for Metroid put us far enough into the game that this plot point had already been revealed -- you start off playing in plain clothes and no armor, and it's clear that it's a woman.
Even as late as 1995, I don't recall any spoiler-phobia about a shocking twist in Super Metroid where one of the aliens who you'd been destroying in the first game now comes to your aid right as you're about to be killed, sacrificing itself for you. Back before video games attempted pathetically to imitate movies, though, no one played them for the narrative, so spoiler panic was ruled out for that reason alone.