There are a couple of reasons why on a subconscious level the TV ad you are mentioning, and in used in that video, the ad known as "Peace Little Girl (Daisy)" ad was important.
At the time of the 1964 elections don't forget what kind of atmosphere the U.S. was in, regarding the horrors of nuclear war. There were two ways to show this, one was from a film that would later be released about concern for nuclear war and the other would be from a film from the time period (perhaps you've seen these):
1. The one from the time was Peter Sellers' 1964 classic movie, "Dr. Strangelove or How I Learned to Stop Worrying and Love the Bomb".
2. And the more recent film (from 1998) that can explain that fear of the time would be the film taken from a Kinky Freeman's book (starring Christopher Walken, Brendan Fraser, Sissy Spacek and Alicia Silverstone):
"Blast From the Past". In which during the Cuban missile crises of October 1962 (so back in 1964, an historically accurate event where the U.S. and the Soviet Union nearly went to nuclear war) the father played by Walken, decided to have his wife, played by Spacek and his young son, played later by Fraser go into their nuclear bomb shelter and then instead of a nuclear bomb, there was an airplane crash on to their property. Very funny movie by the way, but it clearly expresses the worry many people had about nuclear war back in the early and mid-1960's and even as late as the 1980's (with such TV shows as diverse as the original "Star Trek" episodes to ABC's November 1983 movie, "The Day After" addressing and speaking to those fears).
But to the 1964 TV ad in question, "Peace Little Girl (Daisy)", and it's affect on the outcome of the 1964 election. After Kennedy won a close election in 1960, JFK hired a Madison Avenue firm to work on TV advertisements for the Democratic party in 1964, in September 1963. The thing about that was that Madison Avenue had cooled to helping the Democrats after the way Stevenson had run his campaigns in 1952 and 1956 respectively. But the agency JFK had hired in September 1963 was one that had had very effective ads for such companies as Avis: Rent A Car.
Later after JFK had been slain, the company was still working with the Democrats and seen as Goldwater's weakest aspect of his campaigns was how he would lead Vietnam conflict into a nuclear war. So during NBC's Movie of the Week on September 7, 1964 the Democratic party aired the "Peace Little Girl (Daisy)" ad and it made quite impression on the fears that many felt at that time about going to nuclear war. Heck the ad didn't even mention Goldwater by name, but it was enough. The Republican party and Goldwater's campaign complained to the FCC to remove the ad, saying it was misleading of Goldwater's stance and the FCC and the networks (ABC, NBC, and CBS; cable wasn't in existence at that time) complied a few days later, by removing it from primetime. But in doing that, the evening news picked up the ad and what the Republican party and the FCC did and so the ad was now being shown on the evening news for several days and weeks after, and the girl in the ad even made the front cover of TIME. Because of that Goldwater campaign could never recover, and with help from that ad and fears in the U.S. (plus several other reasons) LBJ was able to win the landslide he did.