Definition of the product $\sigma$-algebra

We can write

\begin{align} \bigl\{ \pi_\alpha^{-1}(E_\alpha) : E_\alpha \in \mathcal{M}_\alpha, \alpha \in \{1,2\}\bigr\} &= \{ \pi_1^{-1}(E_1) : E_1 \in \mathcal{M}_1\} \cup \{ \pi_2^{-1}(E_2) : E_2 \in \mathcal{M}_2\}\\ &= \{ E_1 \times X_2 : E_1\in \mathcal{M}_1\} \cup \{ X_1 \times E_2 : E_2 \in \mathcal{M}_2\}. \end{align}

In this form it is clear that this generating set is contained in

$$\{ E_1 \times E_2 : E_\alpha \in \mathcal{M}_\alpha\},$$

and on the other hand, every set in the latter generating family is the intersection of two members of the former, so the two families generate the same $\sigma$-algebra.

$$E_1\times E_2=\pi_1^{-1}(E_1)\cap\pi_2^{-1}(E_2)$$ so is measurable w.r.t.to the first mentioned $\sigma$-algebra.

$$\pi_1^{-1}(E_1)=E_1\times X_2$$ and

$$\pi_2^{-1}(E_2)=X_1\times E_2$$ so both sets are measurable w.r.t. the second mentioned $\sigma$-algebra.

This together proves that the mentioned $\sigma$-algebras coincide.