Let $A$ be a $\sigma$-complete Boolean algebra. By the Loomis--Sikorski theorem, $A$ is isomorphic to $F(X)/M$, where $F(X)$ is the $\sigma$-field of Baire subsets of $X$ (the Stone space of $A$), and $M\subseteq F(X)$ is the $\sigma$-ideal of meagre sets.
So let $h:A\to F(X)/M$ be the isomorphism in question. For every $a\in A$, I have a problem interpreting $h(a)$. Elements of $F(X)/M$ are equivalence classes of Baire sets, so is $h(a)$ the representative of an equivalence class of Baire sets? In other words, is $a\mapsto [h(a)]$?
As Terry Tao explains: $h(a)$ is the equivalence class (modulo the $\sigma$-ideal of meagre and Baire subsets of the Stone space of $A$) of the usual Stone-dual image of $a$ into the Stone space of $A$.
So $A$ is in particular a Boolean algebra, hence it has a Stone space $X=S(A)$ where each element of $A$ corresponds to a unique clopen set $s(a)$ of $X$ (and all clopen sets of $X$ occur as images, and $a \to s(a)$ preserves all BA-operations.
In $X$ we generate the $\sigma$-algebra by the clopen sets, which you call $F(X)$. In $F(X)$ we consider all meagre sets (that happen to lie in that $\sigma$-algebra) and call it $M$ and define a quotient $F(X){/}M$. We can map each $a$ to the equivalence class of $s(a)$ (which is clopen and thus lies in $F(X)$ obviously) so all sets that differ from $s(a)$ by a meagre Baire set; call that class $h(a)$. Halmos in the book shows that all classes of members of $F(X)$ are reached that way (every class has a unique clopen representative which uniquely corresponds to an element of $A$, so it shows $h$ is a bijection) and this $h$ also preserves all countable operations (and of course all Boolean operations too) and so is an isomorphism of $\sigma$-complete algebras.