How do mulligans in a multiplayer game work exactly?
Yes. It's referring to the emphasized part below of rule 103.4:

To take a mulligan, a player shuffles the cards in their hand back into their library, draws a new hand of cards equal to their starting hand size, then puts a number of those cards equal to the number of times that player has taken a mulligan on the bottom of their library in any order.

That number is simply reduced by one.