Re-express A2, B2, and C2 as, respectively,
where simply denote the change in each original quantity.
Your "post-change" expression can then be rendered as
Multiplying out the LHS gives you
Finally deduct your first equation from this last one, leaving
In other words, the change in C is the sum of three influences: The change in A operating on the 'original' B; the change in B operating on the 'original' A; and the "cross product" of A's and B's change.
There's a geometric interpretation of your question...
...think of your original quantity C1 as the green shaded area, it having an area equal to the product of A1 and B1 (with A1 and B1 playing the roles of length and width). Then length and width change by some quantities, resulting in the larger rectangle. Your question amounts to asking what gives rise to the change in the area of the rectangle, and this area-change is the yellow-shaded area. You can see that the yellow area is the sum of three components, each one corresponding to those three 'influences' in my final equation.