Ich bin groß — I am tall.
Ich habe Hunger — I am hungry
How come haben has got the meaning am here?
Does the second line really mean I am hungry or I have hunger?
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As you guessed, the literal translation of the second line would be "I have hunger", while "I am hungry" can also be translated as "Ich bin hungrig".
Both variants are correct and appropriate, although "Ich habe Hunger" seems to be far more used (judged just by my feeling). But I would be surprised if someone would even notice you saying "Ich bin hungrig."
Besides the fact that the first one conceptualizes hunger as something you can have, you can possess and the second one expresses a state of you being, I do not really see a difference in the two german expressions. (And this difference in conceptualisation is just the same in german as in english.)
There is no German word hunger — only Hunger (note the capitalisation). Hunger, the German capitalised word, is a noun; not an adjective. Therefore, it is only logical that you have it as opposed to ‘being’ it. The direct translation of ‘Ich habe Hunger’ is indeed:
I have hunger.
The corresponding adjective is hungrig and it takes sein as you would expect.
Ich bin hungrig.
German is like English in this respect: If you are dealing with a predicative adjective, it must be joined to the subject by the copula sein. If you are talking about a state which is represented by a noun, you need to have or haben.