Why is the area squared and the volume cubed?

Technically, area and volume aren't necessarily squared and cubed. If you're talking about, well, squares and cubes, then this is true. In general, area is some form of length * width, and volume is some form of length * width * height.

But the reason why the units are squared and cubed is because the units effectively get "multiplied" with each other. So for example, if you have a rectangle that is 2 ft by 4 ft, you multiply (2 ft) * (4 ft) = 8 ft2. You always want to make sure your units are consistent with the operations that you perform on the numbers.