Let's $A$ and $B$ are CW-complexes. How to construct CW-complex $A\times B$?
Choose a CW-complex structure for $A$ using cells $e_\alpha$ with attaching maps $\varphi_\alpha$. Do the same for $B$ using cells $e_\beta$ and attaching maps $\varphi_\beta$. Then the products $e_\alpha \times e_\beta$ are cells and the maps $\varphi_\alpha \times \varphi_\beta$ are attaching maps for a CW-complex structure on $A \times B$.
If you are looking for details of a proof, one can be found in Hatcher's book where the statement appears as Theorem A.6.