Programming in Propositional Logic article notation question
The set $X$ is a collection of ordered pairs $[v,i]$ (this is just a strange notation for an ordered pair), in which $v$ is a vertex and $i$ is an index from $1$ to $k$. Informally, $[v,i] \in X$ means that $v$ is the $i$th vertex in the dominating set. The formula $F$ states two things:


There is at most one $i$th vertex (i.e., $X$ doesn't contain $[v,i],[u,i]$ for $v \neq u$).
The vertices in $X$ form a dominating set.


The reason we want the index $i$ is that the logic isn't strong enough to count the size of $X$. It is strong enough to express "at most one" and "at least one", so we use this particular encoding to be able to express "the dominating set has size at most $k$".