You seem to be having trouble with the basic definition of what it means to be an interior point.

Since p is an interior point of A, there is an open set $\displaystyle O\subset A$ with $\displaystyle p\in O$. Similarly there is an open set $\displaystyle U\subseteq B$ with $\displaystyle p\in U$. Then $\displaystyle O\cap U\subseteq A\cap B$ is an open set with $\displaystyle p\in O\cap U$. This shows that $\displaystyle p\in (A\cap B)^o$.