Why "I feel badly" doesn't feel right

What kind of error is it?

It's a grammatical error where the adverb "badly" is incorrectly used instead of the adjective "bad".

Why do people make this mistake?

People often use "badly" due to confusion between how verbs and linking verbs function in sentences, leading them to mistakenly think they need an adverb.

What is correct?

The correct phrase is "I feel bad". "Feel" here is a linking verb and should be followed by an adjective, not an adverb.