Learn Arabic (Modern Standard): من أين أنت؟ أنا مصري.

In Arabic, there is no word for 'am', 'is', or 'are' in the present tense. 'I am Egyptian' is literally just أنا (I) and مصري (Egyptian). You simply put the pronoun and the describing word next to each other!

How do we know if أنت (you) is meant for a man or a woman here?

Without the short vowel marks, أنت is spelled exactly the same for both. It is pronounced 'anta' when speaking to a male, and 'anti' to a female. Because the answer is مصري (a male Egyptian) and not مصرية (a female Egyptian), we know the question was asked to a man.

Why is it من أين (from where) instead of putting 'from' at the end like in English?

In Arabic, prepositions like من (from) must always come before the question word. You cannot leave a preposition dangling at the end of a sentence. So, you literally ask 'From where you?'.

How does the word for Egypt change to mean 'Egyptian' in مصري?

By adding the letter ي (yaa) to the end of a country name, you create an adjective for nationality. مصر (Egypt) becomes مصري (Egyptian male). If a woman were speaking, she would add the feminine ending to make it مصرية.