I am not clear on whether these mathematical objects called fractals get the name from their fractured appearance or from the fact that they have non-integer (fractional) dimensions. Be that as it may, mathematical systems are sometimes associated with fractal objects.
Let's begin by imagining a straight line. It is a one dimensional object. Now if I put a sharp bend in the line it is still a one dimensional object in the sense that each point on the line can be specified by a single number, the distance along the line from some reference point. But the bent line does take up more space than did the straight line. If I bent the line sharply and frequently so that each fold looks like the tooth of a comb, I might even get it to take up most of a rectangle. By repeated foldings of the line it is brought closer and closer to a two dimensional object. Of course more and more length of line is involved as the number of folds increases. In the limit of infinitely many folds, even a finite rectangle would hold an infinite length of line. At that point the folded line has become a fractal object, not truely two dimensional but nearly as efficient in taking up space as a two dimensional object. Perhaps with dimension 1.9 or so.
Another way to think of fractals is as objects with an extremely irregular boundaries. Being from Maine, I am familiar with an irregular coastline. The length of the coastline depends on the length of your measuring stick. There are coves within coves down to the molecular level. As the length of the measuring stick approaches zero the length of a finite extent of coast approaches infinity. This "coves within coves" idea leads to the notion of self similarity. Some fractal objects, like the Maine coast, exhibit self similarity, looking the same at different scales.
Fractals in nature of course break down once the atomic scale is reached. They are only approximately fractal. Mathematical entities are not troubled by the granularity of nature. The true fractals are the mathematical ones.