Math is just a much deeper topic than most others. The things people do in research level math can take a really long time to explain to a lay person because of the many layers of abstraction involved.

It is a very deep and specialised topic. However, there are ways to convey intuition to a 'mathematically mature' audience, and there are quick definitions that are correct but unenlightening. I much prefer the former :)

No, it turns bilinear maps into linear one! If you have three R-modules (one can read K-vector spaces if unfamiliar with modules) N,M,P and a bilinear map N×M→P then there is a unique linear map N⊗M→P compatible with the map N×M→N⊗M which is part of the structure of a tensor product. (What's really going on here in fancy terms is the so called Hom-Tensor adjunction because the _⊗M functor is adjoint to the Hom(M,_) functor, but just thinking about bilinear and linear maps is much clearer)

Ok fair enough, I think I get you.