Let us have two events, $A$ and $B$ whose probabilities are $P(A)$ and $P(B)$. In the absence of any other information, what would be a reasonable probability to assign to $AB$, that is, $A$ and $B$ being true simultaneously?
The first thing that inevitably comes to mind is $P(AB):= P(A)P(B)$. That is, if we have no reason to assume any of the many possible ways in which $A$ and $B$ are dependent, let us assume that they are independent. However, an "independent" seems to stand out as very special among all the possible "dependent" s. Since $0 \leq P(AB) \leq \min(P(A), P(B))$ it seems somehow odd, that one point in this interval would be the "default" somehow.
In which situation would we use one intersection prior over another?