What is meant by an oracle separation between classes $\mathsf{BPP}$ and $\mathsf{BQP}$?
No, an oracle is a black box that solves a problem in a single step. The problem that it solves can be any problem, it doesn't need to be the halting problem.

What Scott is saying is that there is some black box that a BQP machine with it can do more than what a BPP machine can do with it. However, it doesn't mean that without that black box BQP is more powerful than BPP.

A simpler case is P vs. NP. By Baker, Gill, and Solovay's result there is a black box that P with that black box is not equal to NP with that black box. But it doesn't tell us much about what should be the answer to P vs. NP.