It builds not just on algebraic topology, but algebraic geometry, differential manifolds, and category theory (which underlies all three of these). If you don't have at least passing familiarity with the core content of these fields, you're probably not going to find an explanation worth spending any time on.

A better place to start might be de Rham cohomology: http://en.wikipedia.org/wiki/De_Rham_cohomology