In 1957, the Soviet space program had head start over the United States space program when the the Soviet space program first artificial satellite was place in orbit around the Earth during president Dwight Eisenhower's administration. The next milestone achieved by the Soviet Union, before the U.S., took place when Major Yuri Gagarin made the first manned space flight in 1961. At the time, the U.S. space program was lagging quite behind: not only NASA (the national Space Agency created by president Eisenhower) had failed to place a satellite in orbit, but all the rockets tested as space vehicles had failed to take off.
The clear advantage of the Soviet Union over the United States created a lot of tension among the U.S. public opinion, since the capacity of the Soviets to launch vehicles into space also meant that they could easily develop the capacity to build rockets, or missiles, able to carry nuclear warheads across the globe and onto U.S. territory, whereas the U.S. program's failures portrayed the U.S. incapable of deterrence and/or retaliation in case of nuclear warfare.
When presiden John F. Kennedy vowed a promise to get the U.S. space program to beat the Soviets on the race to the Moon, not only was he taking into account the civilian ramifications, such as peaceful scientific research, but also the political and military ones, such as showing the world that the U.S. had the upper hand in case of a nuclear showdown. "The others" [who intend to win, too] Kennedy refers to and challenges are the Soviets and their space program. Even though president Kennedy was assassinated in 1963 and would live to see the fist man, a U.S. man, on the moon, his promise came true to the letter since the feat of a U.S. mission traveling to the moon was made in 1969, "in this {1960s} decade."