Dangling and misplaced modifiers are discussed at length in usage guides partly because they are very common and partly because there are many different kinds of them. But it is not necessary to understand the grammatical details involved to grasp the basic principle: words or phrases which modify some other word or phrase in a sentence should be clearly, firmly joined to them and not dangle off forlornly on their own.
Sometimes the dangling phrase is simply too far removed from the word it modifies, as in "Sizzling on the grill, Theo smelled the Copper River salmon." This makes it sound like Theo is being barbecued, because his name is the nearest noun to "sizzling on the grill." We need to move the dangling modifier closer to the word it really modifies: "salmon." "Theo smelled the Copper River salmon sizzling on the grill."
Sometimes it's not clear which of two possible words a modifier modifies: "Felicia is allergic to raw apples and almonds." Is she allergic only to raw almonds, or all almonds -- even roasted ones? This could be matter of life and death. Here's a much clearer version: "Felicia is allergic to almonds and raw apples." "Raw" now clearly modifies only "apples."
Dangling modifiers involving verbs are especially common and sometimes difficult to spot. For instance, consider this sentence: "Having bought the harpsichord, it now needed tuning." There is no one mentioned in the sentence who did the buying. One way to fix this is to insert the name of someone and make the two halves of the sentence parallel in form: "Wei Chi, having bought the harpsichord, now needed to tune it." If you have a person in mind, it is easy to forget the reader needs to be told about that person; but he or she can't be just "understood."
Here's another sentence with a dangling modifier, in this case at the end of a sentence: "The retirement party was a disaster, not having realized that Arthur had been jailed the previous week." There is nobody here doing the realizing. One fix: "The retirement party was a disaster because we had not realized that Arthur had been jailed the previous week."
Using passive verbs will often trip you up: "In reviewing Gareth's computer records, hundreds of hours spent playing online games were identified." This sort of thing looks fine to a lot of people and in fact is common in professional writing, but technically somebody specific needs to be mentioned in the sentence as doing the identifying. Inserting a doer and shifting to the active voice will fix the problem. While we're at it, let's make clear that Gareth was doing the playing: "The auditor, in checking his computer records, identified hundreds of hours that Gareth had spent playing online games."
Adverbs like "almost," "even," "hardly," "just," "only," and "nearly," are especially likely to get stuck in the wrong spot in a sentence. "Romeo almost kissed Juliet as soon as he met her" means he didn’t kiss her -- he only held her hand. True, but you might want to say something quite different: "Romeo kissed Juliet almost as soon as he met her." The placement of the modifier is crucial.