Learn Russian: Судья решил начислить каждой команде по одному очку. - The judge decided to award one point to each team.

This is the distributive use of по (meaning 'each'). In Russian, when expressing 'each' with the number 1 (as well as 5 and above), по requires the Dative case. So the neuter noun очко becomes очку, and the numeral одно becomes одному. If the judge had awarded 2 points, it would be по два очка, using the Accusative.

Is каждой команде also in the Dative case because of по?

No, it is Dative for a completely different reason. It is the indirect object of the verb начислить (to award points to each team). In this sentence, we happen to have two Dative phrases back-to-back, but only одному очку is triggered by по.

What exactly does начислить mean compared to other words for 'give' or 'add'?

Начислить specifically means 'to credit', 'to accrue', or 'to calculate and award'. You will see it mostly in official, financial, or numerical contexts, such as crediting points in sports, calculating interest in a bank account, or adding a bonus to a salary.

Why do we use одному instead of одной?