Learn English – What does Donald Trump’s “go-to argument” mean

"Going rogue" has a peculiar resonance with a political party (like the U.S. Republican Party) whose symbolic animal is an elephant—namely, the historical connection between "going rogue" and "rogue elephant." Here is the definition of rogue elephant in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

rogue elephant n (1859) 1: a vicious elephant that separates from the herd and roams alone 2: one whose behavior resembles that of a rogue elephant in being aberrant or independent

Roger Courtney, Africa Calling: The True Account of the Author's Strange Workaday Experiences in Kenya, Uganda, and the Belgian Congo (1935) [combined snippets] describes the phenomenon of elephants going rogue:

Often enough a lone animal will go 'rogue.' One reason for going 'rogue' is tuskache—a super-toothache. I remember one such animal creating a reign of terror over a whole district, smashing senselessly through shambas, destroying right and left. Woe betide any person or thing that stood in his path. At any time of the day or night there would come to the terrified natives a sudden squealing and high trumpeting, and the maddened monster would come thundering upon them out of the bush. The hunters employed to hunt down this dangerous beast frequently came across holes where the animal had thrust his aching tusk into the ground. Elephant often dig in the ground with their tusks, searching for roots, but the frequency with which this one did so, also the fact that it was damp, cool soil which he favoured, left no doubt that he was seeking to relieve the agony in his tusk. When eventually he was run to earth it was found that, because of some injury sustained long before, one of his tusks was badly decayed high up.

The idiom "go rogue" is fairly recent in U.S. English. For much of its career, it had a rather unsavory connotation, as this brief parenthetical from SSCP Systems Security Certified Practitioner Study Guide and DVD Training System (2003) indicates:

Documentation, while often boring, serves a very critical purpose. One of the criteria most commonly found in security audits relates to the existence of documentation. If the administrator who set up all of the security were to leave the company or go rogue (become malicious), someone else will need to take over their job.

And from James Stefanie, The Charters Affair: Being a Reminiscence of Dr. John H. Watson (2000):

"Unfortunately, not every part of the body functions properly and in good relation to the other parts all the time. A cell may go rogue and infest the blood with toxins; another cell may grow madly and tumors sprout. As a doctor I am charged with taking the whole of the healthy body and protecting itself from one of its members gone wrong. Does this mean to me that the whole body is evil because one part becomes malevolent? Or does it mean I, as a physician, must keep the body from becoming victim to itself?

But at some point in the past eight years, "going rogue" ceased to be exclusively the province of elephants with aching tusks, and malicious computer security administrators, and toxic body cells, and (perhaps) U.S. Army colonels overseas who have left the (military) reservation in troubling ways. It has become in its new sense, as Rathony's answer observes, a badge of honor—an indiscriminate rampaging with a higher purpose, as it were.

Sarah Palin's book Going Rogue: An American Life (2009) was surely the turning point in the adoption by a segment of U.S. society of the idea that going rogue might be a good thing in a society where, presumably, everything likely to get stepped on is rotten and deserves to be trampled. Hence the new "beholden-to-no-one maverick" sense of "going rogue" that Rathony mentions.

Would Sarah Palin and Donald Trump benefit from pain-free dentistry? I doubt it. In any case, for the time being, the idiom "go rogue" has two highly incompatible meaning: on the one hand, "go wild and put a lot of innocent people at risk of harm"; and on the other, "escape the control of a bunch of cynical handlers and moneyed elites who normally suppress the truth, control the terms of public discourse, and promote the status quo."

The "left and right" part of the quotation, I believe, is not meant to indicate that Trump is veering leftward and rightward (politically speaking) in unexpected ways, but that he is pressing the attack in all directions. Idiomatically, "doing something left and right" simply means doing something vigorously or in all directions, such as "knocking them down left and right."

In a comment, Hot Licks wrote:

It is, of course, a Trumpism to a large degree. "Big league", in the US, refers primarily to "major league" baseball teams and games, or at least to adult baseball, as opposed to "Little League" children's baseball. The idiom has been detached from that meaning over the years, though, and now might be used to refer, eg, to a company which has grown to the point that it can compete with others in a national market. Trump abuses it to a degree to simply mean "big", or "bigly" (sic), as you suggest.