Why do professional writers hate adverbs, and what should be used in their place?

Why do professional writers hate adverbs, and what should be used in their
place?

In response to the death of Elmore Leonard the New York Times has posted a
list of writing tips he composed back in 2001. Among them is the
following:
To use an adverb this way (or almost any way) is a mortal sin.
This is not the first time I have read of professional writers
discouraging the use of adverbs. While technically (adverb!) appropriate
English, why are they treated with such disdain, and what makes a good
adverb substitute?