Learn Turkish: Senin başın neden ağrıyor?

In Turkish, the possessive suffix on başın (your head) already tells us whose head it is. Adding the pronoun senin (your) is optional and is used here for emphasis, like saying "Why does your head ache?"

How does the word neden relate to our lesson about çünkü?

Neden is a question word meaning "why" or "for what reason". It is the natural pair to the conjunction çünkü (because). If someone asks you this question, your answer explaining the cause will likely include çünkü.

Is ağrı a noun or a verb here?

It is a verb! The root ağrımak means "to ache" or "to hurt". To say "is aching", we use the present continuous suffix -yor to make ağrıyor. (Note that ağrı can also be a noun meaning "pain", but Turkish uses the verb form for body parts hurting).

In English we usually say "Why do you have a headache?" Why doesn't this sentence use a word for "have"?

Turkish does not use "have" structures for physical aches. Instead of saying "I have a headache", a Turkish speaker literally says "My head aches" (Başım ağrıyor). So, this sentence naturally asks "Why does your head ache?"