So my question is why do so many people (Mark Twain and Stephen King,
for instance) express such loathing of adverbs and adjectives? Can you
provide some good examples of when adjectives and adverbs are
Count me in the loathing camp. @SF. and @jschabs have already given great answers, though I'll add one additional point: an excess of adjectives and adverbs sends a message to the reader that you don't trust their own imagination. You, the writer, leave nothing to chance, crafting each image and feeling down to the last detail. I think that's rather presumptuous. I'd rather give readers some credit by giving them a general sense of what's going on, a light touch, a nod in the right direction, and let them use their own imaginations for the rest.
Try this analogy on for size: I like to think of verbs as the engine of sentences and nouns as the fuel. The adjectives and adverbs are the paint and chrome of the sentence-as-vehicle, but too much of either just add dead weight. You can write an interesting and meaningful sentence without adjectives or adverbs, but try doing that without nouns or verbs!
For some comparison examples, I'll start with this classic one from Robert Heinlein's Beyond This Horizon. It's not even a complete sentence, it's just a clause: "the door dilated". That's a classic example because it shows how one noun and one verb can speak volumes about where, or rather, when we are. He could have written, "the overlapping panels of the circular door opened radially," but what useful information do all those extra words add? None.
Here's another example, with a full break-down. Compare this sentence:
Adam slogged slowly through the frigid, gelid, and blindingly white
drifts of snow, piling quickly up to his beleaguered hips.
Adam slogged through snowdrifts as high as his hips.
The verb slogged conveys effort, and thus, a slow speed. The reader isn't going to be imagining Adam sprinting around an Olympic track. There's no need to beat him over the head with the adverb slowly. Similarly, the reader already knows snowdrifts only exist in cold places, so frigid and the redundant gelid don't add anything either. If snow isn't white, there had better be a good reason why (as in, relevant to the plot), so that can be omitted. Arguably "blindingly" adds something useful about the visual conditions, but on the other hand, "blindingly" modifies "white", so what about white is so blinding? Wouldn't it be clearer to say, in the next sentence, that the sun, reflecting off the snow, blinded him? Arguably "piling up quickly" gives some sense of the speed with which conditions are worsening, but hang on, now we have a problem. If the "white" of the snow is blinding him, how can the snowdrifts be getting bigger? That suggests conditions where there's no sun visible. That's another problem with too many adjectives and adverbs, you can easily paint yourself into a logical corner trying to juggle so many details.
If you'd like to read more examples of why adjectives and adverbs are loathsome, try any Bulwer-Lytton contest winner, like this one:
The shallow cave behind the mighty river’s thundering waterfall seemed
more like a damp, cold, misty, poorly lit hallway leading from the
shower room in some cheap-dive gym under the Elevated train where mugs
who couldn’t crack the glass jaw of some washed-up palooka on their
best sober day still deluded themselves that they could be somebody;
and yet, Bill thought, “at least it’s got runnin’ water.” — Warren
Blair, Ashburn, VA