It is the side of a square with that area — and more broadly, a balance point between 1 and the number.
The square root of 16 is 4 because a square of area 16 has sides of 4. The same value is also the geometric mean of 1 and 16: the number that sits 'multiplicatively halfway' between them. That is why roots show up whenever growth or scaling is spread evenly across steps.
