can anyone explain what Persist Results in Saved Search does? I've read the SA article and tried it successfully but can't seem to get it's use.

It's for when the search keeps timing out in the user interface

can you elaborate on the specific scenario and how persist solves the issue?

Persist runs search in background, which can be downloaded. If your search takes a long time to run, doing it in background let's it complete when you otherwise wouldn't be able to do it