As a non-fractal object is magnified, no new features are revealed.
As a fractal object is magnified, ever finer new features are revealed.
The shapes of the smaller features are kind-of-like the shapes of the larger features.
The size of the smallest feature of a non-fractal object is called its characteristic scale. When we measure the length, area, and volume at resolution that is finer than the characteristic scale, then all of the features of the object are included. Thus, the measurements at this resolution determine the correct values of the length, area, and volume.
A fractal object has features over a broad ranges of sizes. There is no one characteristic scale. As we measure the length, area, and volume at ever finer resolution, we include ever more of its finer features. Thus, the length, area, and volume depend on the resolution used to make the measurement.
A coastline looks wiggly. You would think that as you enlarge a piece of the coastline the wiggles would be resolved and the coastline would look smooth. But it doesn¹t. No matter how much you enlarge the coastline it still looks just as wiggly. The coastline is similar to itself at different magnifications. This is called self-similarity.
Because of self-similarity, features at one spatial resolution are related to features at other spatial resolutions. The smaller features are smaller copies of the larger features. The length measured at finer resolution will be longer because it includes these finer features. How the measured properties depend on the resolution used to make the measurement is called the scaling relationship.
The dimension gives a quantitative measure of self-similarity and scaling. It tells us how many new pieces of an object are revealed as it is viewed at higher magnification.
Most likely, the statistics that you were taught in school was limited to the statistics of non-fractal objects. Fractals have different statistical properties that may surprise you!