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

Thanks! I think I'll go off and study some mathematics.

We can formally add and subtract simplices. Homology is (very roughly) counting holes using the modulus (%) operator with respect to this addition and subtraction.