Like when compared to what the two major political parties believed in say, the 19th century, todays views seem to be the polar opposite— and the switch seemed to have happened during the civil rights movement. Why did this occur?
The 1960s are popularly regarded as the pivot point where the parties "switched." But this popular perception is misleading for several reasons.
The "switch" had already more or less occurred during the late 1800s. Eric Rauchway has a more detailed explanation of this and the role that William Jennings Bryant and others played in reshaping the parties. So, Republicans were increasingly the party of smaller government by the 1930s, hence opposition to the New Deal. The conservatism of Goldwater and others in the 1960s was a shift, to be sure. However it was more of an evolution and less of a revolution. The top-down use of sweeping legislation and an activist judiciary by JFK and LBJ to address civil rights issues understandably rankled many in the GOP. Goldwater's Conscience of a Conservative gives some insight into these process-based objections and Goldwater's general belief that "libertarianism is the heart of government."
The parties didn't really "switch" in the 1960s, 1970s, and 1980s. It's more that the uneasy Democratic coalition of urban workers, coastal intelligentsia, and Southerners started to fall apart. Responses to civil rights was one factor, but differing economic values and foreign policy also divided the party. LBJ's Great Society represented a massive, unprecedented expansion of the size and role of the federal government. The federal govenment would help raise children, create jobs, spend billions, and take in billions by borrowing and taxing. For some Democrats, even those like Ronald Reagan who'd been sympathetic to the New Deal, this was a step too far. This opposition to the ballooning government favored by the liberal wing of the Democratic party was one force that drove long string of defections by more centrist Democrats in the 1960s and beyond. There was Reagan himself in 1962, then the "Reagan Democrats" of the 1980s. As Reagan said, "I didn't leave the Democratic Party; the Democratic Party left me."
Although the popular narrative is that anti-Civil Rights Southern Democrats all became Republicans, the reality is much more complex. A great many stayed with the Democratic party because they agreed with its economic planks or working-class appeal. Others left the Democrats, but still refused to join the GOP. Some, like Harry F. Byrd, Jr. of Virgina, became independents because of growing discontentment with how their state party was run. Robert Byrd of West Virginia, a former Klansman, stayed a Democrat. And Strom Thurmond of South Carolina became a Republican.