I'm not sure exactly what your question is, so I'm going to interpret it in my own fashion:
Roughly, dimension first arose when Descarte discovered how to coordinatise space. Once you've noted 3 numbers serve to identify a point in space, you can generalise to say, 5 numbers, and that identifies a point in a new space, which is now 5-dimensional.
This seems naive at first, and not really that interesting. Are we just playing games with a list of numbers? But it turns out traditional geometrical terms can be applied to this context. Stuff like curvature, length, angle etc. So it does become interesting.
It turns out that the idea of dimension can be applied to different kinds of spaces and algebras for example topological spaces or rings (a specific type of algebra), these have varying definitions. These spaces or algebras can be finite, so in a sense dimension has nothing to do with having an infinite number of points.
Now dimension as traditionally conceived in space is an intrinsic notion, you don't have to go out of the space to measure it. For example you can conceive of a line in and of itself and not lying in some other space as opposed to say a line that you could draw on the flat surface of a table. A better way to think of the latter is that you've taken your line in and of itself and embedded it (thats the traditional mathematical term) on the table.
You get a fractal when this embedding is complicated in a certain definate sense, that is its self-similar. Fractal dimension measures the complexity of this embedding. So for our example it lies between 1, the dimension of what you're placing - the line, and 2 the dimension of what you're placing into - the plane.
For fractal dimension to differ from one you will need an infinite number of points, otherwise you couldn't hope to get self-similarity as you keep zooming in.
Whats this got to do with Kant I have no idea. I understand he was interested in space and time, and he posited them as forms of our intuition. That is space & time are not out there, but in us: we view the world though space & time spectacles. I wasn't aware that he was interested in the idea of dimension, perhaps you can point out the passage where he descibes it?