Answer: Because many times adverbs signify laziness on the part of the writer. We've all done it. We want to emphasize something, we want to make sure the reader gets what we mean. But we need to find creative and original ways to do that.
Example: Harry looked around lazily. Without Adverb: Harry lifted one eyelid and let his gaze drift around the room.
And why especially dialogue tags?
Answer: Because the reader should know from your writing (other dialogue, body language, internal thoughts, etc.) how the sentence was said. And don't be lazy with your descriptions either. I'm guilty of it too. She rolled her eyes. He grinned. Etc. Push yourself to do better. You won't regret it.
I do want to point out that sometimes adverbs are not evil. But you can do an "ly" search in your document to find out if you've overused. Weigh each one. Take it out and re-read to see if you need it. Chances are it will sound cleaner without.
Any other reasons to avoid adverbs.