I got into a huge argument with my profesor whether the sentence should read: "The clothes smelled bad" or "The clothes smelled badly." I thought it would be "badly," but if I'm wrong I want to know why. My prof's on explanation was that he was right because he's the professor. Please give me the answer and explain why. Thank you!!

I'm afraid, he's right.

(1) My smells badly.
(2) My dog smells bad.

In the first example, the dog is bad at smelling things. The adverb (badly) describes how the verb (smells) is being performed.

Here is a better example:

This apple looks badly. (< This is clearly wrong. It's the same.)

Another common error is correcting people who say, "I am good". There is nothing wrong with this sentence. Many will tell you it should be "I am well". In this instance, "well" is the adjective, meaning "healthy" not the adverb of "good".

I hope that makes sense.

At first I was tempted to say "badly", but, when I thought about it, I would say "bad", is the right choice.
Here is the explanation:
The clothes smell badly, means that tyhe closes have a bad sense of smell. Badly is an adverb defining a verb: smell
The clothes smell bad acts as an adjective defining a noun, the clothes.
That is the reason in my opinion.