A key feature of the Nisnevich topology is that as a cd-structure (cf. [Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies]) it is complete and regular. This implies what Lurie calls Nisnevich excision in DAG XI. The proof of this "excision" relies on the existence of a "splitting sequence" (cf. [Morel-Voevodsky, A^1-Homotopy Theory of Schemes, Lemma 3.1.5]) for any given Nisnevich covering.
Def.:(MV) A splitting sequence for a covering family $\{p_{\alpha}:Spec(R_\alpha)\to Spec(R)\}$ is a sequence of closed subsets of $Spec(R)$ of the form $$ \emptyset = Z_{n+1}\subset Z_n\subset \ldots \subset Z_0=Spec(R) $$ such that for $i=0,\ldots,n$ the morphism $\coprod_\alpha (p_\alpha)^{-1}(Z_i\setminus Z_{i+1})\to Z_i\setminus Z_{i+1}$ splits.
This existence statement needs the space which is covered to be noetherian. Lurie drops the noetherian requirement and pays for the splitting sequence, which doesn't come for free any longer. You find the non-affine situation in section 3.1 of the Morel-Voevodsky paper.
However, here we have $Z_i:=V(a_1,\ldots,a_{i-1})$ an and the first condition above says that $Z_{n+1}= \emptyset$ and the second condition gives a splitting onyeah
$$ Z_i\setminus Z_{i+1}= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_i))^c$$ $$= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_{i-1})\cap V(a_i))^c$$ $$= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_{i-1})^c\cup D(a_i))$$ $$= V(a_1,\ldots,a_{i-1})\cap D(a_i)$$ $$= Spec(R[a_i^{-1}]/(a_1,\ldots,a_{i-1}))$$
