Let $(X_i)_{i\in I}$ be a family of sets. For each $i\in I$, what does it mean to "identify $X_i$ with $X_i\times\{i\}$"?
I know that there exists a canonical bijection of $X_i$ onto $X_i\times\{i\}$ given by $x\mapsto(x,i)$.
If two sets are "identified", does this mean that properties true of one are true of the other? For example, if $i\ne j$, then $X_i\times\{i\}\cap X_j\times\{j\}=\emptyset$; if I "identify" $X_i$ with $X_i\times\{i\}$ and $X_j$ with $X_j\times\{j\}$, can I say that $X_i\cap X_j=\emptyset$?
Concretely, what can I do–which I couldn't do before–after I identify two sets?
You just have a bijection, say $f_i$, between $X_i$ and $X_i \times \{i\}$, defined by $f_i(x) = (x, i)$. But you can't draw any conclusions on $X_i \cap X_j$, because it would involve two different bijections, $f_i$ and $f_j$.
A typical use of this construction is the definition of the disjoint union $X$ of a family $(X_i)_{i \in I}$ of sets. Indeed, $X$ can be defined as the set of ordered pairs $(x, i)$ such that $x \in X_i$ and then the injective map from $X_i$ to $X$ defined by $x \to (x,i)$ allows you to identify $X_i$ with a subset of $X$.