In a lot of ways, using adverbs can be like taking a shotgun and shooting your manuscript with it at point-blank range. And that's why everyone goes around saying, "Don't use adverbs!"
Everyone would be wrong.
There's nothing actually wrong with the adverb. If you learn how to use them effectively.
However, I'll give you the top three ways people like to use adverbs, and, maybe, you can see why there is the temptation to banish them:
1. in dialogue tags
The problem with using adverbs in dialogue tags is that they frequently become a way to tell the audience what is happening rather than to show the audience. For instance, you might use, "he said quietly," when "whispered" would be better. Or you might say, "he leaned close and said into my ear," which would be even better. Or you might say, "she said excitedly," which is just telling me that she was excited, but "she squealed and jumped up and down before saying" shows me that she is excited, which is far more, well, exciting. Basically, adding those adverbs to dialogue tags can be a lazy way of getting around showing what's going on.
And, of course, using descriptive dialogue tags distracts from the actual dialogue, which you don't want to do. The dialogue tag should fade into the background as much as possible, which is why we don't want to draw attention to them by tacking adverbs on. There are few "rules" of writing I believe should be followed unilaterally, but the one about keeping dialogue tags to a simple "said" is probably the one I believe in most. Heck, I think if I could get away with not using them, I would (which is kind of odd considering how much I dislike Hemingway for that very reason).
2. to create redundancies
Unfortunately, the other way we want to use adverbs is to reinforce verbs that don't need reinforcing. In effect, we make a redundant word combination. We like to say things like "he ran quickly" or "she screamed loudly" or "he whispered quietly." We don't need adverbs in any of those circumstances. If he's running, we know that he's doing it quickly, and, if she's screaming, we assume it's loud. In most cases, there are better verbs to replace those combinations anyway, like "he sprinted" or "she shrieked" or "he murmured."
3. really and very
Yeah, people really, really like to use these adverbs very, very much. Frequently, these cause writing to become boring due to word repetition, and, as with any word, you don't want to use them too much. Usually, there are better words.
With all of this going against the adverb, it can be difficult to see legitimate uses for them. It's rather like taking out a sub-machine gun to hunt a deer. The only good reason for that is if you want to save some time in making venison burgers. The trick is knowing when to use the adverb gun and which adverb gun works best.
Personally, I like the adverb as an adverbial phrase. See what I did there? "Personally" is an adverb. In point 3, so are "frequently" and "usually." In point 2, I have "unfortunately." Adverbs in those positions are useful and give a clearer meaning to the sentences. And that's the catch, when we're going to use adverbs, they should provide a clearer meaning; they should provide more focus. Not the same focus. You don't want them to just re-say what you're already saying.
Another good use of the adverb, which provides a greater meaning to the sentence but is not as an adverbial phrase, is to contrast the word you're modifying. Going back to the examples I used earlier, you could have "he ran haltingly," which provides a completely new dynamic to that sentence. You could also say "she screamed hoarsely" or "he whispered loudly." Those adverbs are useful and good and provide new depth to what is being said.
All of this to say that, although I understand the temptation to tell people "don't use adverbs," it's a better solution for people to learn how to use them effectively. There's no real reason to deprive authors of the adverb tool just because some people use them incorrectly. It's not the same as when I was a kid and was given a tool kit one year and proceeded to use the hand drill
something like this
to woodpecker the furniture. Needless to say, it got taken away. However, it would have been better if my parents had taken the time to teach me what to use it on.