The first part is Bannon speaking, informally.
"Go to the mattresses" is a phrase I know--and most people know--only from a novel about the Mafia, "The Godfather," by Mario Puzo. In this novel, Mafia families are about to conduct a war, and to prepare for it they leave their homes, rent apartments, and put down mattresses to sleep on. In any case, it means "to get ready for a serious fight." In the novel, when one Mafia family is planning to fight another, the leaders say "We're going to go to the mattresses."
I don't know what he means by "They set up an outside shop and then Bill and Hillary never mentioned it again."
"They ground through it" means "they got through it with determination and hard work."
"Starr had them dead to rights" means "they were guilty, Starr had the evidence, and Bill Clinton should have been convicted." "They got through it" means that Clinton succeeded in escaping conviction.
The second part is the writer, Michael Wolff, talking about Bannon and Trump.
"Bannon was back" means that Bannon had regained a position of influence, Trump liked him again, Trump was again listening to him "It was not clear what being back meant" is explained in the following sentences.
"Trump being Trump" means that the author, Wolff, thinks Trump has specific, unusual personal characteristics. "Was this true rehabilitation?" raises the question that Trump might appear to like Bannon again, while secretly hating him ("feeling even deeper rancor.") In Wolff's opinion, another person might sincerely forgive Bannon, but "Trump being Trump," it was more likely that Trump held a grudge, continued to be angry at Bannon.