Learn Finnish: Miksi pääsi on kipeä?

It is made of the base noun pää (head) and the possessive ending -si (your). Adding -si to the end of a word is the standard way to show that it belongs to "you" (singular).

Don't I need to use the word sinun (your) before the noun?

Not necessarily! In standard written Finnish, when you use the endings -ni (my) or -si (your), you are allowed to drop the pronouns minun and sinun. The word pääsi all by itself perfectly translates to "your head".

If I wanted to answer "my head is sore", how would that look?

You would just swap the -si (your) ending for the -ni (my) ending. The sentence would be Pääni on kipeä.

Why don't we use the Finnish "to have" structure (sinulla on) to ask "Why do you have a headache?"

Finnish doesn't use a single verb for "to have a headache". You can use the "to have" structure by saying Miksi sinulla on pää kipeä? (literally: Why at you is head sore?). But making the body part the subject of the sentence, like in Miksi pääsi on kipeä? (Why is your head sore?), is just as common and natural.