That's true only if the water gathered by the area in question is all pooled into a smaller area and has its depth measured there.
For example if you had a hillside of a certain area acting as a catchment which all drains into a pond of a tenth the area, then if an inch fell on that hillside, the pond would end up with 10 inches more water in it.
The "an inch of rain" measure doesn't refer to the pond, but to the hill, on the ficticious assumption that it doesn't drain away, but all stays in situ until you've had a chance to measure it, and then slops down the hill.