Why was the Soviet development of satellite so significant to the people of the United States?
The Soviet Union put satellites (and people) into space before America did, which seemed unthinkable!
We Americans have been rigorously trained to think of ourselves as the best and most important country in the history of the world. By some benchmarks, like military and economic power, we are indisputably number one, and have been since World War II. This encourages a degree of yahooism and jingoism, which is often accompanied by an unattractive strain of anti-intellectualism; this has held America back from other benchmarks, like the sciences.
20th Century America was home to some of the world's greatest scientific minds--Albert Einstein, Nicola Tesla, Enrico Fermi, Werner von Braun and Edward Teller, to name five. It didn't produce many of these top-tier scientists, though: all five of these men were born and educated in Europe, mostly in countries we would go (or had gone) to war with.
When the Soviets launched Sputnik in 1957, America was officially on notice: We needed fewer football heroes and more science education.
We still lag behind much of the world educationally, but we did get a man to the moon before the Russians did (though they got a man into orbit before we did, and got a woman into space decades before us). Sputnik jolted America out of its complacency in this particular regard.