OOO I wrote a paper on this! I am not a 'historian', however, just a post-highschool student with love of history, so take what I say with a grain of salt.

Although there was certainly some technology-races prior to Sputnik 1, you could say 'things got serious' with the 1957 launch of the Sputnik sattellite. This was followed only one months later with a much larger Sputnk II sattellite which was carrying Laika, the first organism in space (which is also about the size of an atomic bomb - what a coincidence!).

After these events, we can see a considerable effect on the American populace: Time magazine warned the Russians could “put a powerful atomic bomb on the moon”, and journalist Eric Savareid predicted “Russia will soon stand astride the world, its military master”. This was around the time where much of the public saw the USSR as an equal, if not more advanced than the US, when really the US had much more military and economic power. In fact, a Gallup poll showed that 39% of Americans beleived - mistakenly - that "Russians could wipe out most cities in the US in a matter of hours".

It can be argued that this lead directly to the education reform in the US that happened in the late 1950s. Curriculum changes occurred almost across the board, and the ‘National Defense of Education’ act tripled funding for the National Science Foundation. You see, just before the Sputnik launches, American education had followed 'progressive' policies: family living and practical arts over sciences and math. Sputnik reversed these policies, driving a rapid reprioritization of science, math and engineering, so that America could compete against the Soviet space program.

So, with the forming of NASA in 1958, the 'space race' began in earnest. For the American public, this was one of the most 'visible' parts of the Cold War, where most of the competition happened behind closed doors, so it was a big deal. America's first attempt at a Sputnik-esque missile was an enourmous failure, with the Vanguard malfunction, which added even more shame to the American space program.

The rest of the space race up to the moon landing was truly a technology and Public Opinion race. The USSR puts the first astronaut into space, Yuri Gagarin, in 1961, followed by the US doing the same not even a month later with Alan Shepard. Kennedy then acknowledges the race with his famous promise, "I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the Earth," which also politically acted as a challenge to the Soviet space superiority in the late 1950s. Politically speaking, the space race's main value was in the Public Opinion victories to be gained, but also the military possiblities - which Eisenhower certainly recognized in Sputnik. This is why NASA was such a big deal, as it was separated from military programs, which had governed the space race until its formation.

Finally, landing on the moon itself clearly showed that America had overtaken the Soviets, and it showed this Globally with over 50 million people watching the origional broadcast. In 1970, a few months after the lunar landing, Soviet dissident Andrei Sakharov wrote in an open letter to the Kremlin that America's ability to put a man on the moon proved the superiority of a democracy. While this was a letter written by a radical, it mirrored some of the Russian attitude shifts.

Many historians consider the American victory in the Space Race as a key victory in Deutscher's 'Great Contest' between Communism and Capitalism.

EDIT: If you want to know more, or want sources for anything specific, just ask!