Warning: I'm a
graduate student at the
University of Chicago Divinity School. In other words, I am way too interested in this topic, and an aspiring
academic.
The Rapture, in the form discussed in this node, is a relatively recent development in Christian theology--you will note that Webster 1913's writeup under rapture doesn't make any mention of people getting carried off to heaven. It may have been floating around England in the early 19th century, but the first person to clearly state the idea was John Nelson Darby, an English theologian and early leader of the Plymouth Brethren.
The idea of the rapture stems from Darby's system of dispensationalism, which reconciles the contradictions in the Bible by explaining that God has made different covenants with different groups of people, each of which is still valid. Of particular importance to Darby was the promise that God made to King David in 2 Samuel 7:12-13--namely, that one of David's descendants would rule over a kingdom that would last until the end of the world. From a very early date, Christian writers had interpreted this as a reference to the Church established by Jesus (who, according to the Gospels, is a descendant of David). Darby, on the other hand, said that the Bible did not equate David's kingdom with the Church--and therefore he concluded that Jesus would return before the final end of the world to establish a kingdom on earth.
The problem is that the kingdom promised to David was a Jewish kingdom in which the Jews would be the center of the world. What would happen to all the people who found salvation by grace through Christianity? Darby hit upon 1 Thessalonians 4:16-17 to support the idea of a rapture, when all the Christians (and other saved non-Jews) would be swept up to heaven, getting them out of the way so that the Jews could rule the world after the Second Coming of Christ.
The story of how this belief became so popular in the United States is a complicated one, which will have to wait for another writeup. But it is safe to say that many American Christians who believe in the Rapture are not fully aware of the theological background. But it is important to note that Darby did not develop the doctrine rapture because he thought Christians should be spared from suffering with the infidels; he actually taught that many people who suffered through the Tribulation would eventually be saved--and that the suffering would be followed by a millennium, a 1000-year long Golden Age.