Take "don't use adverbs" as an example. There's actually a second half of that sentence that goes unspoken way too often: "Don't use adverbs if you can use a stronger verb." But even with that last part, it isn't a full piece of advice – it's been trimmed down and compressed too much. To really understand it, you gotta look at both the context and the reason for its popularity, which are usually bound together.
This advice owes much (if not all) of its modern popularity to Stephen King. In his memoir/craft book, he writes:
“I believe the road to hell is paved with adverbs, and I will shout it from the rooftops. To put it another way, they're like dandelions. If you have one on your lawn, it looks pretty and unique. If you fail to root it out, however, you find five the next day... fifty the day after that... and then, my brothers and sisters, your lawn is totally, completely, and profligately covered with dandelions. By then you see them for the weeds they really are, but by then it's—GASP!!—too late.”
So, essentially, he's saying that you should be conscious of how and when you use adverbs so that they don't take over your writing like dandelions taking over a lawn (let's ignore the anti-dandelion sentiment for a moment). That's pretty different from "Never use adverbs" or "adverbs are bad", right? And that's still only one quote from a whole ass book that revisits and expands on many, many interconnected pieces of advice (because no piece of writing advice works in isolation).
To unpack this even further, look at the sentence "He walked quickly to the cupboard and picked up a cup cautiously, his hands shaking nervously." It presents three main problems:
First problem is the repetition of the 'ly' sound: paragraphs of that will give your writing a dull, repetitive quality. This will kill any rhythm or flow that it might otherwise have.
Second problem is one of redundancy: "hands shaking" tells readers this character is nervous well enough on its own without "nervously". (This actually introduces a second second problem, which is that the more words in a sentence, the more a reader has to parse, so redundant words can make that experience unnecessarily tedious or difficult.)
Third is one of focal point: every adverb adds or suggests information, but when there are loads of them one after the other, readers will stop really seeing them – so they won't focus on the information that's most important. To me, the most important adverb in that sentence is "cautiously": even though this character is rushing and nervous, he's still taking the time to be cautious about what he does. That could show readers a lot about his character or his situation.
So, I'd probably rewrite as "He hurried to the cupboard and cautiously picked up a cup, his hands shaking." There is still an adverb, but it's a load-bearing adverb; removing it would strip a layer of meaning from the sentence. Someone else might rewrite as: "He [hurried/walked quickly] to the cupboard and picked up a cup in his shaking hands. Even in his haste, he was cautious." This kind of thing depends a lot on stylistic preference and how much emphasis you want to put on certain information – but either way, this is a much more helpful and productive approach than ruthlessly pruning all adverbs, ever.
As for why it's so popular, well. Stephen King is a wildly successful author. That gives his advice a lot of weight, and for good reason! He speaks from decades of experience as a professional writer, so you can be sure it's sound advice. But his fame and credibility also mean that people are more likely to swallow whatever he says without taking the time to understand it. Those people then share the advice in regurgitated form, usually paring it down into those pithy one-liners because they're memorable. Other people share this form because "everyone else" says it's good advice (and, let's be real, because the one-liners look good on social media graphics), and this continues on and on ad nauseam until the real substance of the advice is lost.
