Fiber of morphism homeomorphic to $f^{-1}(y)$

The required equality $p^{-1}(\operatorname{Spec}B_i \cap \operatorname{Spec}B_j)=\operatorname{Spec}(B_i \otimes_A k(y)) \cap \operatorname{Spec}(B_j \otimes_A k(y))$ follows from the following:

Given a fiber product $(X\times_S Y,p_X, p_Y)$ of two schemes $X$ and $Y$ over $S$. Then for any open $U$ of $X,$ the fiber product $U\times_S Y$ of $U$ and $Y$ over $S$ is isomorphic to $p_X^{-1}(U)$.

Applying this to the above situation gives the desired result.