I won’t get into much detail here other than to talk about a few example of why you would want to calculate vector cross products and briefly explain what they are. Wikipedia has a great entry on them which will explain all the math behind them.
Given two vectors, for instance, two sides of a triangle, the cross product will give you a vector that points perpendicular to those vectors. This has many uses in 3D animation. You can calculate surface area (1/2 the magnitude of the cross product of vectors of two sides of a triangle will give you the area of that triangle). You can calculate surface normals, which show which direction the polygons of your surface are pointing. You can use cross products to align sprites to the camera or cull geometry that is pointing away from the camera. One thing I used cross products for recently was in a particle setup where I needed to create an aura around a character. I was able to figure out which surfaces of the geometry were pointing perpendicular to the camera and use that as a basis for particle emission. So, many uses.
Technical Repository 
I really enjoyed reading this article, keep up making such interesting posts!