Many natural objects have parts that are similar to other parts of it, like the fern. This is just the beginning of the usefulness of fractals. Many phenomena like heart beat, turbulent weather, the stock market, and mountains also exhibit self-similarity. Other great example of fractals are landscapes. The picture below is a mountain generated using fractals.
Fractals are excellent at creating realistic images, because they can model natural objects well.
Most fractals are also infinitely detailed. From the animation below, you can see that the complexity never disappears if you zoom into the right place. This fractal does not have true self-similarity, because the smaller versions are not exactly the same as the larger one. Generally though, fractals preserve the detail of an object, no matter how far you zoom.