Suppose you want to build a system that using some M features of the current weather (humidity, temperature, pressure etc.) will predict whether it will be sunny in 3 hours or not - to this end you manually collect a dataset of N measurements (samples) of those features together with the outcome 3 hours after taking the measurement. This dataset can be used to find a function that will map the current features of weather to a real variable from 0.0 to 1.0 - the likelihood of it being sunny in 3 hours given the samples we collected. Of course, since we cannot manually collect measurements of all possible combinations of features, we want to extend the domain of this function beyond just the combinations of features that occured in our measurements - we want to "generalize". The values of the function for those "generalized" inputs are our guesses about how the likelihood function behaves for measurements other than the ones we have taken. The guess for a given measurement is much more likely to be correct if we at least have _similar_ measurements in our collection.
Now, if there are too few features (M is too low), it might not be possible to do a good prediction. The basic aspect of the "curse of dimensionality" is that as the number M of features grows (linearly), the number of all possible combinations of the features grows exponentially and hence the number N of measurements you need to do to get a good representation of the whole space of possible feature combinations also grows exponentially. So either you have to collect much more measurements, or the function that is inferred will have to "guess" much more often, without having good evidence from the dataset in the form of similar cases.
What I refer to as "all possible combinations of features", can be thought as a high-dimensional space and hence it is customary and sometimes useful to use geometric language in analysing the problem.
Now, if there are too few features (M is too low), it might not be possible to do a good prediction. The basic aspect of the "curse of dimensionality" is that as the number M of features grows (linearly), the number of all possible combinations of the features grows exponentially and hence the number N of measurements you need to do to get a good representation of the whole space of possible feature combinations also grows exponentially. So either you have to collect much more measurements, or the function that is inferred will have to "guess" much more often, without having good evidence from the dataset in the form of similar cases.
What I refer to as "all possible combinations of features", can be thought as a high-dimensional space and hence it is customary and sometimes useful to use geometric language in analysing the problem.