Learn French: Il a affirmé que la vitesse avait provoqué les dégâts. - He stated that the speed had caused the damage.

This is reported speech. The original statement was likely La vitesse a provoqué les dégâts (The speed caused the damage). Because the reporting verb is in the past (Il a affirmé), the original passé composé shifts one step further back in time to the plus-que-parfait (avait provoqué).

Why is les dégâts plural when the English translation uses the singular word 'damage'?

In English, 'damage' is usually an uncountable, singular noun. In French, physical damage or destruction is very often expressed in the plural as les dégâts. You will also frequently see it used with the verb faire, as in faire des dégâts (to cause damage).

Can I leave out que and just say Il a affirmé la vitesse... like we do in English?

No, you can never skip que in French the way you can sometimes drop 'that' in English. It must always be included to connect the main reporting clause and the reported speech.

Does a affirmé mean 'affirmed'?

They are related, but affirmer is used much more commonly in French than 'affirm' is in everyday English. It simply means 'to state', 'to assert', or 'to maintain' something confidently as a fact.