What is the point of the "respect" requirement in cut property of minimum spanning tree?
I don't have the book handy. I assume that


"safe" means that $(u,v)$ can be added to $A$ and we get another subset of a minimal spanning tree of $G$, and
the goal is a correctness proof of Prim's algorithm¹.


Now, say the cut did not "respect" $A$. That means that there already is an edge in $A$ that crosses it. By choosing a second edge from that cut -- which may happen even with pairwise distinct weights! -- we may introduce a cycle and lose the tree property.



By requiring that the cut "respects" $A$ we know that $A \subseteq S$ and we can not introduce a cycle².

The assumptions of this theorem may be stronger than strictly necessary -- as long as we do not create a cycle, we do not per se need to care about other edges in the cut -- but we can build an algorithm around them, so the theorem is a convenient way to ensure its correctness.




Note how the theorem directly implies that fact since Prim's algorithm maintains such a cut and chooses light edges from it to grow the tree.
In the algorithm, we'll always have $A=S$.