There are different ways of representing $B$. One possibility is to let $B$ consist of all closed subsets of $2^\kappa$ of positive measure; another possibility is to allow all positive Borel sets. Two conditions are equivalent if their symmetric difference has measure zero.

Each basic clopen set has a finite set of coordinates. By ccc, each open set is almost the same as some open set that uses only countably many coordinates. So every closed/Borel set $p$ is equivalent to a set that uses only countably many coordinates. I guess that these are the "coordinates of $p$".