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 with . Similarly there is an open set with . Then is an open set with . This shows that .
Here is a way to think about both the interior of a set and the closure of a set.
The interior of a set is the ‘largest’ open subset of the set.
The closure of a set is the ‘smallest closed set containing the set as a subset.
Here is another suggestion. If you can find the small book Elementary Theory of Metric Spaces by Robert B Reisel, then that would be a great introduction for you. It is written as a self-study or Moore-style class.
p is an interior point of both A and B, or
p is a boundary point of A and or B.
If p is an interior point of A & B, there must be a neighborhood of p containing only points common to A and B, and this neighborhood becomes the neighborhood of p in A^B and so is an interior point of A^B.
If p is a boundary point of, say, A, every neighborhood of p will contain points in A and not in A and so will have some points in common and not in common with the same point in B. Thus p is a boundary point of A^B.
Therefore the only interior points of A^B will be common interior points of A & B: Ao^Bo.
I cannot give a simpler explanation.