Can anybody make an attempt at explaining the content of "Axiomatic Characterization of Ordinary Differential Cohomology" further than the article did?
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.
We can formally add and subtract simplices. Homology is (very roughly) counting holes using the modulus (%) operator with respect to this addition and subtraction.
I don't know any algebraic topology; the idea of
https://en.wikipedia.org/wiki/Homology_%28mathematics%29
looks extremely clever to me but I can see where this subject matter would start to get challenging to explain.