One uses the following trick.
By the projection formula we have $${p_2}_*(\mathcal{O}_{X\times Y_1}) \cong {p_2}_*(p_2^*M_1^{-1} \otimes L_1) \cong M_1^{-1} \otimes {p_2}_*(L_1)$$ and since $X$ is complete (and $k$ is algebraically closed) it follows from the Künneth formula that $${p_2}_*\mathcal{O}_{X\times Y_1} \cong \mathcal{O}_{Y_1}$$ which gives the desired isomorphism on $Y_1$ by uniqueness of inverses up to isomorphism in the Picard group.
In case I misunderstood and you were asking what the version of the Künneth formula is in algebraic geometry there is an explanation here at the Encyclopedia of Mathematics (search for algebraic geometry to get to the relevant part).