Approximation to a second order partial derivative
I think that this question would be more suitable for math.SE, since it is purely mathematical and not specific to economics.

In your notation, the first derivative would be given by
$$
\partial_A v_i = \frac{v_{i+1} - v_i}{\Delta}.
$$
In order to compute the second derivative, we need to apply this formula with $\partial_A v_i$ itself in place of $v$. This gives
$$
\partial_{AA} v_i = \frac{ \frac{v_{i+2} - v_{i+1}}{\Delta} - \frac{v_{i+1} - v_i}{\Delta}}{\Delta}.
$$
A bit of algebra reveals that this is the same as
$$
\partial_{AA} v_i = \frac{v_{i+2} - 2v_{i+1} + v_i}{\Delta^2}.
$$
Now if your grid is fine enough, you can lower the index $i$ on the right-hand side without changing the result significantly. This gives the equation that you came across. It is generally preferred due to the "symmetry" that it displays between $i-1$ and $i+1$.

For more on this topic, see e.g. Wikipedia at finite difference.