Why is the surface area of a three-dimensional figure measured in square units rather than in cubic units

Because when a three dimensional object is folded flat, its area is calculated, and then it is no longer has volume, only length and bredth. Volume has cubic units.

It's a translation.

A reflection across a line is like flipping that object like a pancake over the line.  It gets turned upside-down.

When you flip the pancake again, now across a parallel line, it gets turned right-side up again, just in a new location.