We have P(A OR B) = P(A) + P(B) - P(A and B) = P(A) + P(B) since A and B = null event and P(Null event) = 0 (since P(Not Null) = 1). We also have independence to be defined as P(A|B) = P(A) and P(A and B) = P(A)P(B).
Recall though that since A and B is the null event, then P(A and B) = 0 regardless of what A and B are. This violates the assumption of independence since P(A|B) = P(A) will be strictly positive and depend on the actual event (instead of being zero).
The other intuitive thing is that independence means that knowing one thing about another variable means no advantage when it comes to knowing something about the variable of interest. Since we know that both variables are disjoint, this implies that knowing something about the other directly tells us about the variable of interest since they are disjoint.
There are a few ways to look at this but hopefully the above has given you further insight.