The definition of $A\times B$ is given to be a condition on pairs. Let $a, c \in X$ and $b, d \in Y$, then $(a, b)\sim_{A\times B}(c, d)$ if $a\sim_Ac$ and $b\sim_Bd$.
I have shown that this is an equivalence relation but want to give an explicit description of the equivalence class of $\sim_{AxB}$.
If given a two sets $X$ and $Y$ with equivalance relations $\sim_A$ and $\sim_B$ respectivally then the equivalance relation $\sim_{A\times B}$ is an equivalance relation on $X\times Y$ where $(x,y)\sim_{A\times B} (x',y')$ if and only if $x\sim_A x'$ and $y\sim_B y'$.
An explicit description
With this framework you can view an element $(x,y)$ as the class $(x^*,y^*)=\{(x',y'): (x',y')\sim_{A\times B} (x,y)\}$ and view $A\times B $ as the set of all these classes. Note there is no overlap since $(x_1,y_1)\sim (x_2,y_2)$ means $(x_1,y_1)\in (x_2^*,y_2^*)$ and $(x_2,y_2)\in (x_1^*,y_1^*)$ .
An example
Let $X=\mathbb{Z}=Y$ and $A=\{r: n\in\mathbb{Z}, n=r\mod{2}\}=B$, then $A\times B=\{(0^*,0^*),(1^*,0^*),(0^*,1^*),(1^*,1^*)\}$.
Where $x_1$~$_A x_2$ means $2| (x_1-x_2)$ and $y_1$~$_B y_2$ means $2| (y_1-y_2)$.
So in this framework $(1^*,0^*)=\{(2m+1,2k):k,m\in \mathbb{Z}\}$