Boy, those folks over at Everything2.com are just sharp as tacks, aren't they? That explanation or definition or observation or gloss, or whatever it was meant to be, is about as useful as a chocolate teapot.
In statistics, the curse of dimensionality just refers to the fact that the sample size needed to address a given type of problem satisfactorily increases exponentially with the number of variables under study. Thus, if the response you are interested in only depends on a single variable X, which can take values from 0 to 10 (for example), you might take 10 samples equally spaced along that range and feel that you had covered it reasonably well. Add another possible variable Y, which can also take values from 0 to 10 - to sample the (X,Y) range with the same fidelity would require 100 samples. In general, for n variables under study, 10^n samples.
This gets to be a problem, e.g. in microarray studies, where the RNA expression levels of up to 8000 genes at a time can be measured. Trying to find where the action is in that 8,000-dimensional space of genes can be tricky.
Mathematically, this results from the exponential dependence of the volume of the unit hypercube on dimension, and the difficulty of studying all regions of the space adequately.
And if I have allowed a note of detectable pique to creep into this commentary it's because I have never read quite such an idiotic characterization of statistical reasoning before in my life ("problems that map an input to an output" seems to cover more or less any cognitive process) and my professional sensibilities are (slightly) offended.
What sionnach doesn't know is that in his jalapeno-sodden absence at the colourful end of Montezuma's digestive tract, Natasha has revealed all. She has coughed up ... not just furballs. She has rolled over ... not just to have her tummy-wummy rubbed. She has spilled the beans ... not just in a hazmat-spat-catfight with Boris on the beanbag. AND she confirmed that as cats age 9 times faster than humans (not to mention with greater aplomb), she was dragging microarray studies on to the doorstep while sionnach, a mere fox in kitty clothing, was still bleating for someone to make kind comments about his oddly-pointed ears.