Vector space identity from Chow's "You Could Have Invented Spectral Sequences"
The first relation: take any vector space with subspaces $C, Z\supseteq B$. The natural surjection $Z/B \to (Z+C)/(B+C)$ has kernel $(Z\cap C +B)/B$ which is isomorphic to $Z\cap C /B\cap C$ so the isomorphism you want follows from the first isomorphism theorem.
$\partial^0$ is the map formed as the direct sum of the maps $C_{dp}/C_{d,p-1}\to C_{d-1,p}/C_{d-1,p-1} $ given by $c + C_{d,p-1}\mapsto \partial(c)+ C_{d-1,p-1}$