There are probably deeper answers to this, but one that comes to mind is that just because our world (high level physics considerations aside) is three dimensional, that doesn't mean that the only things that we want to measure in it are three dimensional. As an example let's assume you have a data set, say a point for $N$ different people and associated to each person you have a number $R$ of numerically-valued characteristics, e.g. height in centimeters, weight in kilograms etc. Then to each person we can associate a point in $R$ dimensional space, whose coordinates are the values of the characteristics.
Given this set of data we'd like to be able to tell whether two people are 'close' in some sense, so we need a metric, a notion of distance between the points. We already have the Euclidean metric in 3 dimensions, and we can generalise it to $R$ dimensions, so that's one notion of distance that we can choose. There are other metrics of course, many better suited, but this is an example of a situation where we need a metric on a space with more than 3 dimensions to tell us something about the real world.