If the shape of the plates of a capacitor used a fractal shape then their surface area would increase dramatically. Surface area is the one of the most important aspects of a capacitor.
Imagine the two plates to be colored yellow and green, and the black line to be an insulating layer.
David Hilbert found this curve in 1892. It was one of the first space-filling curves ever found. The first level is made by dividing a square into quarters, the joining their four centers to make the initial curve. Next, if you divide each quarter into quarters and do the same to them -- with the small change of rotating the first and last curves so they can be easily joined to their neighbors -- you get the next level. If you keep doing this for smaller and smaller squares you end up approaching what is called a space-filling curve. But what we are interested in is the length of the line that is the curve. In the base example, the line is 3 units long.
The first step up in complexity, the line has 15 segments, but each segment is smaller. When you measure it, the curve there is 7.5 units long -- half a unit short of 8 units long.
They keep increasing at an accelerating rate, each time approaching a power of 2 in length: 32, 64, 128... At just the sixth, the curve has 16,383 segments and is almost 256 units long.
In another six iterations the curve will be almost 16,384 units long. Another six iterations after that it will be more than a million units long!
The amazing thing is that the size of the surface grows, but the size of the package doesn't.
Now, this example is just an illustration of a line filling out a flat, two dimensional plane, but an actual capacitor should be imagined as a flat, two dimensional surface which gets deformed up and down, left and right, back and forward, so that it gradually fills a 3D volume. The membrane ends up with an extraordinarily large surface area as the complexity grows.
It's similar to how our lungs are fairly small organs considering they have a surface area equivalent to a membrane about 8 meters by 8 meters in size. Or how all the blood vessels in your body would measure about 100,000 kilometers long, if placed end-to-end. Such systems are remarkable.