I have read the Wikipedia definition of cap product. Here is the Wikipedia definition that I know. https://en.wikipedia.org/wiki/Cap_product
What I understand: For a good space, for example, an orientable closed $n$ manifold using the Poincare Duality and definition of cup product, I can convince myself that there should be such a map, but I don't understand what this bilinear map actually mean. Can anyone please clarify the definition? I just want to know what motivates us to define cap product, and what actually it means.
Thank you very much.