I have heard that given two sheaves $A$ and $B$ on a variety, one can identify elements of $Ext^d(A,B)$ with complexes of sheaves $$0\to B \to C_1 \to \cdots \to C_d \to A \to 0.$$
My questions are,
How do I see that this is true?
If I have obtained an element of $Ext^n$ by some other method, can I explicitly construct the $C_j$ sheaves and the differentials?
I am sure this is well-known, so I'm marking it also as "reference-request".