Note that, when we scale an object up, we're increasing <em>all of its lengths</em> 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 = <em>10 units</em>. 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 <em>both of its lengths </em>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 <em>we're multiplying twice, once for each length</em>.
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 <em>cubed</em>, because we're scaling each of the cube's <em>3 lengths </em>by that factor.
