Learn French: Tu as quel âge ?

In French, you don't be an age, you have an age. To ask about someone's age, you literally say "You have what age?". This is why we use as (from avoir, to have) instead of a form of être (to be).

Why do we use quel instead of a word meaning "how"?

Because the French sentence is literally "You have what age?". Quel means "which" or "what" when it is paired with a noun (like âge). Since French doesn't use the adjective "old" to ask this, a word meaning "how" wouldn't make sense here.

Why are the question words (quel âge) at the very end of the sentence?

In everyday, spoken French, it is very common to keep the standard statement order (Tu as) and simply put the question words at the end. A more formal way to ask this is Quel âge as-tu ?, which brings the question words to the front and swaps the subject and verb.

Why is it quel and not quelle?

Quel is a question word that has to match the gender of the noun it describes. Because âge is a masculine noun, we use the masculine form quel. If you were asking "what" about a feminine noun, you would write quelle, although both forms sound exactly the same.