As a matter of fact, the "dimension" is a precise (and delicate) notion that has a mathematical definition (in fact,several). Maybe you know what the dimension of a vector space is (which is an integer). For Cantor subsets, one obviously needs a different definition, that would account for their different self-similarity.

Maybe your book says how it defines dimension. Anyway, you can find lots of ressource on the internet: look for "fractal dimension". In the case of your subset, you can cut it in 2 parts that look like the initial subset up to a scale . Similarly, if we cut a line segment (d=1) in 4 we get parts that look like the initial subset to a scale , if we cut a square (d=2) in 16 we get smaller squares like the first one to scale 1/4, and if we cut the cube (d=3) in we get smaller cubes like the first one to scale , and in dimension , we can cut a cube into smaller cubes that look like half the initial cube up to a scale . So for the middle-half Cantor subset, you would have , hence . This is not a proof, since I gave no definition of dimension, but it gives a reason why it should be .But the book say that it is 1/2-dimensional.