The physics idea can be thought of as a generalization of electromagnetism. You can describe how electrons interact with electromagnetic fields by re-interpreting the vector potential in terms of the theory of connections on principal bundles, in this case U(1)-bundles, and that can be generalized to groups besides U(1), which leads to non-Abelian gauge theory. It turns out that the electromagnetic tensor can be thought of as a kind of curvature. If you study differential geometry, you learn that you can push around vectors on a surface by parallel transporting them, and that curvature measures the path-dependency of where the vectors end up if you push them along from point a to point b. With principal bundles, this is generalized to things other than vectors, like group elements. The Yang-Mills Lagrangian of the standard model is built out of this sort of generalized curvature. This is all fine for mathematicians, like myself, but the actual physics version generally makes our heads explode because you have to quantize the theory and it gets kind of ridiculous. In a lot of the math side of gauge theory, you end up just using the classical field theory and study the critical points of the action to come up with really weird mathematical facts.
I have no idea why U(1) x SU(2) turns out to be appropriate for electroweak forces, but I can give the somewhat trite answer that it's justified by experiment (and I think that may even turn out to be the "official explanation" by physicists to some degree--I'm not sure how deep of a justification there is beyond that it works).
The cross product is a fairly simple construction where you just take one group and put it in the first slot and the second group in the second slot and just think of it as a group where each slot acts the same way it normally does. It's interesting that such a simple construction, involving two very basic groups, like U(1) and SU(2) (aka a circle and the unit quaternions, respectively) would turn out to be the key to describing two of the fundamental forces of nature. Another interesting point from a physics point of view is that, contrary to what you might expect, the U(1) in the product turns out not to correspond to the U(1) of the electomagnetic force--it's actually a different copy of U(1) that lives inside the product. Or at least that's what physicists tell me.