There were Southerners promoting leftist ideas long before the Cold War Era, like the "Bloody Harlan" strikes in Kentucky, the "Mine Wars" in West Virginia, The Loray Mill Strike in North Carolina, and the Alabama Communist Party in the 30s.

Earlier in the 20th century Kentucky folksinger and labor organizer "Aunt" Molly Jackson sang:

"I was raised in Kentucky, / Kentucky born and bred, / but when I Joined the Union / they called me a Russian Red".

If I may I'd like to link a previous answer of mine with additional sources about these pro-Union protest songs from Harlan County, Kentucky which used traditional Appalachian music and lived experience to develop class-conscious critiques of the coal mining system. Many of them include notions of racial equality similar to what other commentors have mentioned regarding the Civil Rights movement.

See also:

KELLEY, ROBIN D. G. Hammer and Hoe: Alabama Communists during the Great Depression. Twenty-Fifth Anniversary Edition ed., University of North Carolina Press, 2015.

Romalis, Shelly. Pistol Packin' Mama: Aunt Molly Jackson and the Politics of Folksong. Urbana: University of Illinois Press, 1999. Print.

Though I'm not an expert on Robbins specifically, the song came out in 1966 so we can get a sense of what he was likely referring to by looking at what was going on in the South around that time.

What was going on was the Civil Rights movement. 1966 is after the passage of the landmark Civil Rights Act of 1964 which banned, among other things, discrimination in employment and schools on the basis of race, banned discriminatory application of voting registration rules, and discrimination in places of public accommodation. This was one of, if not the, biggest single actions to undermine Jim Crow and open racial segregation in the United States.

The "brand new way of life" Robbins was referring to was likely a way of life where the South could no longer force black people to sit at the back of the bus, couldn't force them to use worse drinking fountains and accommodations, and had to either educate their children in integrated schools or (as some chose to) close down public schooling altogether. As you can imagine, a culture that is centered around explicit white supremacy is greatly disrupted when it can no longer legally be so open about its discrimination. And, as you can imagine, the passage (and often enforcement) of federal anti-discrimination laws was pushed by people who were opposed to the system in place in the Jim Crow South who tended to not be southerners (or, at least, respected southerners). [As an interesting aside, the famously racist Alabama Governor George Wallace got his start as a civil rights lawyer helping black people. It was after he saw that opposing black people's rights is the single most important political position in the South that he became so rabid].

So that's the "way of life" that is new and which the south considered to be trouble - one where they had to at least pretend to treat black people somewhat equal to white people.

As to your question about left-wing activists traveling to the South to promote left-wing ideologies, the answer there is 'sort of, depending on how you define things'. The ideology that was being pushed onto the South was of racial equality. As to the traveling promoters of the idea, the most likely reference there is to the (very visible at the time) Student Non-Violent Coordinating Committee. These were largely younger, educated kids from northern universities who first rose to attention by participating in the Freedom Rides in 1961 where integrated buses came to the South following the Supreme Court decision desegrating interstate travel. They were met with violence, public opposition, but also cameras and reporters, publicizing the struggle. The late Congressman John Lewis (a true American hero in my opinion, and a person whose presence stayed with anyone who'd ever met him) was part of the SNCC and you might be familiar with some of their work from the discussions of his contributions after his passing last year.

The SNCC continued to be involved in the South including by running black voter registration drives throughout the rural South, and challenged existing political party structures, upending the status quo which had existed. It led marches throughout the South, including in Selma Alabama, and it helped organize quite a few black organizations including one which would become the Black Panther party.

The SNCC also had the same tensions as the larger civil rights movement between the non-violent integrationist wing and a more militant one, with those tensions being resolved in favor of the latter in the summer of 1966 with the election of Stokely Carmichael to SNCC leadership.

So that's the background to what was going on when Robbins put out this song in 1966. It was after more than half a decade of very public, escalating conflict in the south between civil rights activists and defenders of the Jim Crow status quo. The conflict seemed to only be getting worse and I could imagine some skeptical folks would have looked at an SNCC which started out preaching nonviolence and integration but by 1966 was striking out into a more militant direction as the logical culmination of the civil rights movement. Charitably construed, Robbins' message may have been that the left-wing Northerners who stirred up black people in the rural South and preached integration and an end to discrimination talked about how it would help things but, by 1966, the changes had caused division (as whites committed to segregation refused to give it up) rather than increased harmony. The activists referred to were most likely the SNCC and those like them.

As to your final question on - did these activists have an impact on the public opinion - the answer is again 'sort of, depending on what you consider 'public opinion' and 'impact'. They upended the established Southern system of discrimination and challenged white supremacy where no credible challenges seemed to exist before. That had a huge impact. It had an impact on the white people who supported white supremacy (if you asked them, probably a very negative impact) and on those who were opposed to it. If you're asking if they changed people's minds to be more left-wing - yes they did, but not the minds of most southern whites. The freedom rides drew attention from people who were more or less 'on the sidelines' of civil rights. Northern whites who didn't really care about what happened in the South, or who vaguely supported it on the mythology that it was for the best, were disabused of their notions by the extreme violence meted out against even white university students who were following the law. That, in turn, led to the changing opinions of 'neutral' whites, as well as hope for those (white and black) who were fighting against discrimination. However, public opinion of whites in the South was not immediately changed by the rides, as can be seen by the extreme opposition to integration throughout the South. If anything, it hardened against change as that change was forced from the federal level.

I fear that's a too-condensed summary, but it's hard to talk about that time in the US without all the nuance leading to a treatise. Hopefully that gives you some areas to look at further.