How do you visualize the classification of data with many more features than three? You could arbitrarily pick two-three of the features and graph the points on those dimensions but which ones to pick is a very hard question. So we would rather an algorithm pick the best way to represent the data that has some nice theoretical and computational aspects.

That is why statisticians use principal component analysis (PCA) to create visuals (as well as solve many more important problems). Basically PCA will pick the best three meta-features for you. By meta-feature I mean a combination of features. Geometrically PCA is rotating the dimensions to capture the most variability. Then it will collapse the data perpendicular to that feature and pick out the next best meta-feature in the same way as the first. It will do this for as much as you want.

Below is a rotating three dimensional picture of data classied by a Support Vector Machine algorithm. The data really has about ninety dimensions but I used PCA to pick the top three combinations of dimensions so I could graph it.

Rotating three dimensional PCA picture to understand the data better

Though 3D pictures can be helpful, it is better to show several 2D pictures so that what you see doesn't depend on the rotation. Here is one example below.

Trying to visualize high-dimensional classification