Hand-wavey non-math details: Imagine a wall with illuminated colored shadows positioned randomly on it. The colored shadows are created by semi-transparent co

Hand-wavey non-math details: Imagine a wall with illuminated colored shadows positioned randomly on it. The colored shadows are created by semi-transparent colored balls some distance from the wall. Each set of balls of the same color are the same distance from the wall (e.g. the red ones are 1m back, the blue ones are 2m back, etc). Now, you are just looking at the wall. You don't know there is anything not on the wall (i.e. the higher dimensional objects projecting the pattern on the two dimensional surface). You just see colors interspersed seemingly randomly. In order to make sense of all the two dimensional randomness, you can up-project the points to a higher dimensional space (i.e. try to re-create what projected them down in the first place). That's the kernel trick. You're projecting lower dimensional data into a higher dimensional space. The goal of all this is to make your data linearly separable (i.e. so you can draw a line (2d) or plane (3d) or hyperplane (4d+) between you classes (colors) of data). You can't separate the colors when they are random points on the wall, but you sure a heck can separate the colors when you see the colored balls grouped by individual color at specific offsets from the wall. That's a very simple 2d to 3d projection where you would draw a plane to separate your colors. For more interesting classes of data, you'll probably end up in much higher dimensional spaces.