Suppose X is distributive lattice having $x_{1},x_{2},x_{3}........x_{n}$ elements and
Suppose Y is distributive lattice having $y_{1},y_{2},y_{3}........y_{m}$ elements
then total number of elements in X $\times$ Y = n$\times$m
A non distributive lattice does not satisfy the distributive property because it contains 2 or more pair of elements having common LUB and GLB.
So if X and Y are distributive lattice
then if $x_{i}$ and $x_{j}$ have GLB $x_{k}$ and LUB $x_{l}$ and
if $y_{i}$ and $y_{j}$ have GLB $y_{k}$ and LUB $y_{l}$
then in Lattice X $\times$ Y,
elements $\left ( x_{k}, y_{k} \right )$ and $\left ( x_{l}, y_{l} \right )$ can be GLB and LUB of only 2 elements $\left ( x_{i}, y_{i} \right )$ and $\left ( x_{j}, y_{j} \right )$
this applies on every pair of Lattice X $\times$ Y. Hence Lattice X $\times$ Y holds distributive property and hence is a Distributive Lattice.