I'll attempt to answer your question as best I can, but I'd like to ask from what year during the Civil Rights movement that poll was taken, as it would make a considerable difference. The movement wasn't a single event, but a process that lasted over more than a decade, with the introduction of new leaders and personalities throughout. Anyway, I'll break my response up into what I believe are the main reasons for its success.

All the way with LBJ

The assassination of JFK on November 22, 1963, was a historic moment for the United States, for more than one reason, but for the purpose of the question, it's key for one main one in particular; it allowed for the passing of legislature. In the year leading up to his assassination, JFK had announced his intentions to pass civil rights legislature, which made sense given his progressive and likeable style. However, there has arguably never been a politician more hated by the right (except for perhaps B. Clinton), and, had it not been for his assassination, it would have been near impossible for him to have passed anything through congress, and we wouldn't have seen anything even close to the landslide that was the 1964 election. It was this carrying of JFK's popularity that allowed for a highly sympathetic congress which passed practically anything the stetson wearing Texan threw at them, including the Civil Rights Act of 1964. Had things not worked out the way they had, the US Civil Rights Movement would have almost certainly be slowed down.

A decade of acronyms

Now, let's talk about MLK Jr. Admittedly, not everyone agreed with the most well known promoter of the Civil Rights Movement, as Richard Nixon once put it, he'd be "Tough on crime", which basically meant that he was against civil rights protest. People viewing protesters as criminals was a strong undercurrent amongst conservatives, but on the other hand, many of those in the centre were also weary of, what was to them, the Supreme Court, as well as congress, handing out rights left, (right), and centre. Not to mention, in the mid 60s we see the rise of Malcom X, who later goes on to become El Hajj Melik El Shabazz, along with other groups and people, but I'll touch on that later. In any case, MLK Jr was seen by many as the rational, reasonable, and controlling voice of the movement, who stopped it from getting out of control. I mean, imagine being an average American and looking at MLK's peaceful approach towards obtaining rights, and then turn around and see Malcom X talking about creating a sovereign nation within the US, just for black people, and openly talking about violence. As such, MLK's ideas were more readily accepted, due to the clear cut down the middle, with 'radicals' like X on the other side. Were it not for individuals such as Malcom X, H Rap Brown, Stokely Carmichael, and many more, making openly aggressive statements and denouncing MLK's approach, MLK would have himself seemed more like a fringe personality.

Those are the main two I can come up with for now, but please feel free to ask any further questions, as more reasons may come to mind with time. Hope that helps!