How exactly does curse of dimensionality curse? In what way does the curse of dimensionality affect the predictions?
I know that as the number of predictors increases the observations that are geometrically near decrease, so we have to spread out more to capture the nearest neighbours.
So my question is , in what other ways, than being computationally expensive , does the curse of dimensionality cause the non-parametric model to perform poorly?
 A: The idea of nearest neighbours is that, due to continuity, other points close to your point of interest have values close to the value of your point of interest. If you have to spread very far out to find the 100 (for example) closest points, well then these points are not very close or neighbours anymore, and thus there's no reason that their values are relevant at all to help you predict the value at your point of interest!
A: The curse is that a lot of of things that work in lower dimension don't scale well (i.e. grow/shrink too fast compare to linear) with dimension. E.g. a measure of data quality is more than 2 times worse when you double the number of dimension. The curse comes in many form: the range of the data, the density of the data, the distribution, etc.
Take the example from ESL book. Says in every 1 dimension the possible values are in $[0,1]$, your data covers  $[0.1,0.9]$, i.e. the range of data $r=0.8$. That's pretty good of a range if your data is just 1D.
The generalization of range to dimension $p$ is $0.8^p$. As $p$ increase, the range of your data shrinks really fast. In 2D, $p=2$, even though in each dimension, your data spans 80% of possible range, the 'volume' range that it capture is only 64%. In 10D, $p=10$, your data only cover 10% of the space! If you want to cover 80% of the space, in each dimension, your data need to span $0.8^{\frac{1}{10}} \approx 0.98$% of possible range - that's very expensive.
A: It curses mainly in computational sense. However, if you expand the dimensionality from small to large to infinite, asymptotically any point that's not this point becomes infinitely far away. 
If you're at point O, and are looking at points A and B, then both $OA=\infty$ and $OB=\infty$, and you can't distinguish which one is farther.
