Well, when you think about it a line (interval) in has measure equal to it's length while the cantor set has measure zero. It is tiny. Almost every real number is not in the Cantor set. So it sort of makes sense that it is not 1-dimensional.

A line is 1-dimensional, and a line has measure greater than zero.

The Cantor set is not a point, and so really shouldn't have dimension zero.

Does that help