Johnson did a very good job of balancing the image of carrying the torch of President Kennedy while starting to etch out his own stamp on America. Johnson was able to paint himself as a great alternative to Senator Goldwater, whose outspoken opinions on Vietnam seemed far too extreme for most Americans of the time period. Johnson made the "Great Society" a benchmark of his administration, embodying the liberal principles that appealed to much of the bourgeoning idealism of the time period. He was able to capture the public's imagination by wanting to "do more for so many" in the opening days of his tenure as president. Winning by a landslide electoral vote, Johnson was able to use the election results as a mandate for him to be able to do what he set out to do. This allowed him to literally grab Congress by the throat and be able to advance legislation with ease. Johnson was also able to parlay much of this public support into his policy on Vietnam, a move that ended up costing him all of his public good will in only a few short years.
In my opinon, the main way that Pres. Johnson won the support of the American people during his first year in office was to cast himself as the heir to the ideas of John F. Kennedy. The American people showed their support, in part, by reelecting him in a complete landslide less than one year after he first became president.
Many Americans saw JFK as a fallen hero. They came to revere him and his ideas. Johnson took some of these ideas (notably the civil rights legislation that JFK had proposed) and used his legislative skills to push them through Congress. By doing so, he identified himself with JFK's legacy. I think that this is much of what made him popular with Americans.
Lyndon B. Johnson was considered one of the most brillant politicians of his time, and, as stated above, he rode on the romanticized "Camelot" of the Kennedy presidency. Known for his "Johnson treatment," the figurative arm-twisting of powerful politicians in order to get legislation passed, Johnson, indeed, rode into the presidency on the wave of Kennedy's ideals. His "Great Society" included laws that upheld civil rights, furthering what Kennedy had begun. He instituted hiring quotas for minorities; he pushed through legislation on Medicare and Medicaid with the Social Security Act of 1965; and he began a "War on Poverty."
But, Johnson also greatly escalated the war in Vietnam, a war that became increasing unpopular. (Rumour has it that he profited from the war with a bridge company in his wife's name.) Consequently, the president who was so popular in 1964 and 1965 grew increasingly unpopular; in fact, he was despised by many at the end of his term. He withdrew from the campaign for re-election with the growing opposition to the Vietnam War after he had a poor showing in New Hampshire in 1968. After the debacle of Vietnam, Johnson is considered a failure in foreign policy, but he is ranked favorably with some historians for his domestic policies.