Why it is incorrect to say "I am hungry" or "I am thirsty" in Spanish?

Because that would mean you are the actual entity of hunger or thirst.

This is just like saying I am heat instead of I am hot.

In Spanish, sentences are sometimes taken literal while in English they always aren't. Like if you say "I am thirsty" you don't mean you are the literal feeling of thirst. You mean that you have thirst but you are saying "I am".

But in Spanish, you cannot say "I am thirsty" because you are not thirst itself. Instead, you say "I have thirst" and not "I am hungry" but "I have hunger"