How does the Prestige Leader mechanic work?
The instructions in the rules are pretty clear if you read them and follow them directly. If it helps, it pretty much boils down to:


if there's a single prestige leader, and they gained prestige last turn as they became prestige leader or afterwards, they get 1VP and 1 card
if there's a single prestige leader, but they only gained prestige on earlier turns, they get 1VP
if there's a tie for prestige leader, all tied players get 1VP


In practice, the condition for getting 1VP and 1 card (the first bullet point) is pretty much always equivalent to whether the sole prestige leader gained prestige on the previous turn. The only time it's not is if, say, they became prestige leader during Develop, they lost it/became tied during Settle, and they became sole leader again during Consume; that's of course quite rare.

In order to remind yourself of all this, you pass around the prestige leader token. Whenever there's a single leader, they get that token, and they put any prestige they gain on it as a reminder that next turn they'll get 1VP and 1 card. When they get that bonus, they take their prestige back as a reminder that next turn they'll just get the 1VP. When there's a tie, you flip it over to the tie side, and the former leader takes back their prestige, as a reminder that it'll just be the 1VP bonus next turn.

And... the point of this is to provide small ongoing rewards for prestige! In some cases, it's even enough to make prestige a good overall plan for your game.