In learning about the Consistency of Martin's Axiom through Kunen and Jech with help from other set theorists, I have come to a basic question about marrying these proofs:

What is the connection between the nice names argument in Kunen and the boolean valued proof in Jech regarding a limit on the number of partial order-names for a partial order which satisfies the countable chain condition?

I am more familiar with the nice names argument and trying to understand why every boolean value of an element of a set is determined by a countable anti chain in the partial order. I can't see how values in the boolean algebra are boolean sums of the values of an anti-chain in the partial order. Any help to understanding the link would be appreciated.