Imagine a line. That is a one dimentional object—there's only two degrees of movement along a line. Now, square the line and you end up with a plane; a two dimentional object with four degrees of movement—left, right, forward and backwards. Square a plane and you get an object that is three dimentional with six degrees of movement—left, right, forward, backward, up and down.
Fractal dimentions are those that aren't whole numbers like one, two or three.
Now imagine a one dimentional object that curves about in a plane. Technically it's not a line, but something that is made up of segments of lines, all connected yet at no point does this object intersect itself. A straight line has a dimention of one. Put a curve into it, and the dimention goes up slightly, say 1.00000000000001 or so. The more curvy it is, the more it curves about and fills up the plane, the higher the dimention goes, towards two. But it doesn't quite become a plane.
An object with a dimention between two and three is a plane that curves and folds around to define a volume.
But the other kicker is that you can only follow along the line (or plane) and can't move outside it. A being who lives along such a line might not be aware that its universe is kinked that way—for all it knows it lives in a one dimentional universe unaware that it lives in something that is higher than one, but not quite two dimentions.
If not, don't worry about it. You don't need to really know about it.