Like many things, it's largely practice. Adler's How to Read a Book is an accessible and well-regarded work on the topic, and its "three readings" framework, if not to be taken as an exhaustive gospel on the ways of reading—some people "can read" while failing to consistently achieve Adler's first reading, so there's plainly more that could be explored, and the book's not intended as an academic treatise on all the components of reading—is, at least, a solid practical guide.
I think the best explanation of how this all works is to liken it to listening to music: everyone can listen to music. Not everyone understands or appreciates a wide variety of music, or gets as much out of it as some do. One encounters "jazz is just noise" or "hip-hop is just noise", et c., opinions with some frequency—much as one encounters "the 'classics' are overrated, boring crap", when it comes to fiction, for example. A major part of learning to read well is internalizing the process of experiencing new genres or ways of expression, which one may not enjoy at first, just as many experience some discomfort and very little pleasure the first time they deliberately try to broaden their musical-appreciation horizons. This may mean reading texts about the texts to better understand them, or watching lectures, but it also means a whole lot of sheer exposure to the works themselves, just as it does with music.
If there's a single piece of advice I could give, it'd be to ask more questions, more often. Interrogate the text. What's wrong with this assertion, if anything? Does the text go on to address any problems I can see? If it does, is that satisfying? Do others see this problem? If not, what did I miss? Is this confidently-stated premise or postulate reasonable? If not, can the argument stand without it? Why is this character in this story? What's the structural reason this scene exists? If it seems pointless, but this work is very well-regarded, I'm probably missing something—what might that be? Does it have thematic or textural import that I'm failing to spot, and so, perhaps, missing much of the message of the work? Et c.
However, a lot of reading failures I see among Web posters are so basic that all the above is almost too advanced, and I don't know exactly how to address those problems. Simple reading comprehension failures that, incredibly, persist even when pointed out; misunderstanding how human language works, often taking the form of excessively literal readings that become weirdly and unhelpfully adversarial while failing to engage what the text was actually expressing; that kind of thing. This is part of what's behind the extreme over-explaining and frequent shouldn't-be-necessary disclaimers in writing by seasoned Web forum posters—I have a suspicion that readers who are that bad are not, in fact, extremely common, but are just, for whatever reason, unusually likely to engage in Web discussions, leading to the standard, gratingly-poor writing style of Web forums, aimed at pre-empting bad readers from posting useless flames or diversions from the topic, which efforts are still, often, insufficient.
> often taking the form of excessively literal readings that become weirdly and unhelpfully adversarial while failing to engage what the text was actually expressing
I see you, too, have seen people not understanding that Humbert Humbert is not supposed to be liked, and that the whole book is him trying to justify his actions. God, the number of people who misunderstand Lolita because they can't accept anything about it except at face value is way too high.
> I have a suspicion that readers who are that bad are not, in fact, extremely common, but are just, for whatever reason, unusually likely to engage in Web discussions
Sadly, bad readers are that common. I taught at a high school, and even some of the other teachers were bad readers, and most the students couldn't get anything but a very literal interpretation out of any text they read.
Is there a good source to teach me that skill?