It's describing a technique for classifying data into one of two categories (A or B) statistically, based on existing observations of data from categories A and B.
If we could just draw a straight line or plane that separated all of the A observations from the B observations (that is, A and B are "linearly separable"), keeping as far away from both as possible, we could say that any observation that hits the line is 50/50, and anything on one side or the other has a particular statistical change of being either A or B depending on how far away it is. (The fact that it's straight makes the statistics easy.)
But not all data sets can be separated easily with a straight line or plane; sometimes they're best separated with some kind of curve ("non-linearly separable"). The "trick" here is to create a higher-dimensional curved surface -- basically twisting our original space where A and B are -- in such a way that now we can make a straight cut between A and B on the twisted surface. The way our cut intersects the new curved surface corresponds to the best separating curve in the original space. (The hard part is picking the mathematical formula or "kernel" to create the twisted surface.)
So, for example, in the video all of the A points were outside of a particular circle and all of the B points were inside the circle. We create the curved surface above and copy all of the A and B points over. This allows us to separate A from B with a straight plane; its intersection with our curved space corresponds to a circle in the original space.
If we could just draw a straight line or plane that separated all of the A observations from the B observations (that is, A and B are "linearly separable"), keeping as far away from both as possible, we could say that any observation that hits the line is 50/50, and anything on one side or the other has a particular statistical change of being either A or B depending on how far away it is. (The fact that it's straight makes the statistics easy.)
But not all data sets can be separated easily with a straight line or plane; sometimes they're best separated with some kind of curve ("non-linearly separable"). The "trick" here is to create a higher-dimensional curved surface -- basically twisting our original space where A and B are -- in such a way that now we can make a straight cut between A and B on the twisted surface. The way our cut intersects the new curved surface corresponds to the best separating curve in the original space. (The hard part is picking the mathematical formula or "kernel" to create the twisted surface.)
So, for example, in the video all of the A points were outside of a particular circle and all of the B points were inside the circle. We create the curved surface above and copy all of the A and B points over. This allows us to separate A from B with a straight plane; its intersection with our curved space corresponds to a circle in the original space.
See also: http://en.wikipedia.org/wiki/Support_vector_machine