The question asks one to find an equation to describe the change in surface area of stacks of 1 in by 1 in cubes. The first figure is a single cube, the second figure has a base of two cubes with one cube stacked on the left cube, the second has a base of three cubes with the previous figure stacked on top of it (still keeping the right triangle appearance).
My trouble is in finding a way to describe the change in area through an equation.
You're asked to find out an expression for the surface area, why bother with the volume? Anyway, finding the progression of the volume is a trivial problem (just the triangular number sequence).
The surface area is a little harder.
Sketch the diagrams for $n=1,2,3,4,5$. You'll need at least those many cases to be very clear about what's happening. Some cubes have 5 faces "exposed, some have 4, some 3 and some 2. Label them as "5-faces", "4-faces", etc.
Count how many there are in each configuration for these lower values of $n$.
You'll find that for $n \geq 4$,
The number of 5-faces = 2 (constant, since they're the "hanging corner" cubes)
Number of 4-faces = $(n-2) + 1 = n-1$ (a diagonal line at the "exposed surface" plus the single "bottom corner" piece).
Number of 3-faces = $2(n-2)$ (the left and right "edge rows")
Number of 2-faces = $T_{n-3}$, where $T_n$ is the $n$-th triangular number. This is a little tricky to see, but you'll see the 2-faces always stacked in a triangular profile with base $n-3$. So the number of 2-faces is $\frac{1}{2}(n-3)(n-2)$.
Now to find the total area, add all those up, weighted by the areas of each type of face. That is,
$$A = (5)(2) + (4)(n-1)+ (3)(2(n-2)) + 2(\frac{1}{2})(n-3)(n-2) \\ \therefore A= n(n+5)$$
Even though you established this expression for $n \geq 4$, you can quickly verify that it also holds for $n=1,2,3$. Therefore, you have a truly general solution.
You can prove it rigorously by induction by considering the change in the number of each type of face from $n \to n+1$. The reasoning would largely be intuitive, based on visualising the profile of the shape, just spelled out a little more formally.