How does high surface area to volume ratio affect the function of the mitochondria?
More surface area of the inner membrane -> more space to fit things -> more ATP-producing structures can fit -> more ATPs produced per mitochondria = better mitochondria!
The mitochondria's main function is to produce ATPs via structures that sit on its inner membrane (complexes of the electron transport chain (ETC) and ATP synthase). The inner membrane separates the intermembrane space (outside) from the mitochrondrial matrix (inside).
The ETC structures use a sequence of reactions to move hydrogen ions (H+) from inside (the matrix) to outside (intermembrane space) and ATP synthase then harnesses the potential energy and flow of H+ back inside the matrix to produce ATP.
Here you see one complete "set" of these ATP-producing structures:
The more sets of these (ETC + ATP synthase) that can fit on the inner membrane, the more ATPs mitochondria could produce at a time.
Folds (called cristae) increase the amount of surface area of the inner membrane without changing the size of the outer membrane or volume of the overall mitochondria.
This allows more ATP-producing sets (seen below as the 3 yellow circles and 1 black stalk) to fit inside the mitochondria, which enables more ATPs to be produced per mitochondria.
This figure only shows 2 sets to keep the visuals simple but there are actually sets of these covering as much of the inner membrane as can fit: