Learn Finnish: Miksi hän ei puhu? - Why does he not speak?

In a positive sentence, "he speaks" is hän puhuu. But in negative sentences, the main verb changes into a shorter form. For verb type 1 words like puhua (to speak), you drop the final -a or -ä from the dictionary form to get this short stem. That leaves just puhu. You don't double the final vowel like you do in the positive form.

Why is the negative word ei used here instead of en?

In Finnish, the negative word acts like a verb and changes depending on who is doing the action. Since the subject is hän (he/she), we use the third-person form ei. If the sentence was "I do not speak", it would be minä en puhu.

Does hän mean "he" or "she" here?

It could be either! Finnish does not have gendered pronouns, so hän means both "he" and "she". The translation shown uses "he", but "Why does she not speak?" is an equally correct translation.

Does the word order change to make this a question?

No, Finnish is much simpler than English here. English adds helper words like "does" and swaps the word order around. In Finnish, you just put the question word miksi (why) directly in front of the normal statement hän ei puhu (he does not speak).