Learn Swedish: Du är väl inte arg? – Jo, jag är visst arg! - You are not angry, are you?

Because the first person is asking a negative question by using inte. Whenever you want to say "yes" to contradict a negative question or statement in Swedish, you must use jo. It essentially means "On the contrary, yes!"

Where does the "are you?" come from in the Swedish sentence?

It comes from the little word väl. Swedish doesn't use tag questions at the end of sentences like English does with "are you?" or "don't they?". Instead, adding väl in the middle of the sentence shows that you expect a certain answer or are asking for confirmation.

What does visst do in the reply?

visst is an emphasizing word that translates to "certainly", "indeed", or "actually". When you use it together with jo to contradict someone, it makes your answer much stronger. The speaker is stressing, "Yes, I certainly am angry!"

Why does the first sentence start with Du är (You are) instead of the question word order Är du (Are you)?

When asking for confirmation using väl, Swedes often use normal statement word order (Subject + Verb) rather than reversing them to form a strict question. You just let the tone of your voice rise at the end of the sentence to show it is a question.