Note that, when we scale an object up, we're increasing all of its lengths by some factor. Let's think about the effects this has in one dimension first.
If you have a line segment that's, say, 5 units long, and you scale it up by a factor of 2, its new length will be 5 x 2 = 10 units. By definition, a line segment - and any one-dimensional object, only has length, so we're only scaling up one number.
For 2-dimensions, let's think about the area of a square with sides of length 5. Unscaled, this square has an area of 5 x 5 = 25 square units. Now, if we scale this square by a factor of two, we're going to be multiplying both of its lengths by two, getting us an area of (5 x 2) x (5 x 2) = 100 square units. Notice, that if we rearrange this equation to put the scale factors out front, we get 2 x 2 x 5 x 5 = 2² x 5²; our scale factor shows up square because we're multiplying twice, once for each length.
Going up to 3 dimensions, we can look at a cube with edge length 5. Its volume would normally be 5 x 5 x 5 = 125 cubic units, but scaled up by 2, we get (5 x 2) x (5 x 2) x (5 x 2) = 1000 cubic units, which we can again rearrange to make 2 x 2 x 2 x 5 x 5 x 5 = 2³ x 5³ - here our scale factor is cubed, because we're scaling each of the cube's 3 lengths by that factor.
