After reading the article I still have no idea what that "oracle" is thus I cannot assess the "why" consulting it. Even the article doesn't go into much depth into the "why" rather into the "how".
The point is simply to explore the consequences regarding computability and complexity if Turing machines can access such oracles. By definition, a machine with access to the first type of oracle gives you a O(1) solution to any NP-complete problem. But how does it react on more complicated problems, such as NEXP-TIME (non-deterministic exponential time) or PSPACE (polynomial space)? Does it even help or not? The idea is to test different class of problems with different classes of oracles to create analogs of the complexity hierarchy.
The hope is that the relationships between these analogs provide a deeper understanding of the standard hierarchy.
The "why" is that if we assume, as a thought experiment, that we have a magic black box that always gives the correct answer to some question that we otherwise don't (or can't) know the answer to, it can help us come up with and prove results about computational complexity that apply to the "real", oracle-less world too.