Learn Russian: За такие ответы мы обычно начисляем по пять очков. - For such answers we usually award five points each.

The preposition по is used here for distributive quantification, meaning 'each' or 'apiece'. By saying по пять очков, the speaker means five points for every single answer, rather than just five points total for the whole group of answers.

What case is пять in here?

In modern everyday Russian, numbers from 5 onwards usually take the Accusative case when used with the distributive по, which is why we see по пять here. Older or highly formal grammar prescribes the Dative case for 5 and above (which would be по пяти), so you might see that in textbooks, but по пять is standard today.

Why is it очков instead of another case?

Whenever a number like пять is in the Nominative or Accusative case, the noun following it must be in the Genitive plural. Since очко means 'point', it becomes очков. The addition of по does not change this basic number rule.

Why do we use за такие ответы instead of a Dative construction to say what the points are given for?

When giving a reward, penalty, or payment in exchange for something, Russian uses the preposition за followed by the Accusative case. You are essentially exchanging the points for the answers.