The Roots of Anti-Authoritarianism
First, and this will surprise some who know my theological commitments, Luther's 95 Theses in 1517 contributed to an anti-authority attitude. Please understand me here. I am not saying that what Luther did was wrong or that he was himself sinfully trying to undermine authority. I don't think that's true at all. However, his actions did lead many of his countrymen to revolt and to sinfully rebel against the government. Such rebellion was put down, but one could say that this was the first historical example of a society turning in on itself, and the common people achieving a significant degree of rebellious success.
The major contributing factor to the prevailing anti-authority attitude was the Enlightenment, which emphasized as its major principle the autonomous will and reason of man over against the authority of God. Prior to the Enlightenment, every person who lived, essentially, believed that God held the uppermost authority. Post-Enlightenment, however, the European citizen came to view himself as an autonomous being. Thus, the French Revolution, in which the French citizenry did in fact overthrow the reigning government (prompting the celebration of Bastille Day--think about it: should a French Christian celebrate this day?). In addition, the American Revolution represents a revolt against established authority. One cannot swiftly say that this event was wrong, but surely this event represented a negative response to established authority (whether just or unjust). It also gave rise to Modernism, in which man himself is central, autonomous, and uninhibited (purportedly) by sin. Man's reason discovers truth, not the leading of God's Spirit. In pre-Modernism, authority was located in God's Word. Now, it was untethered and subject to the whims of man.
It took many years for Enlightenment thought to ferment. When it did, in the late nineteenth century, it overthrew the authority of the Bible and the Christian church. This began in Europe and spread all over the globe. Men began to realize that truth located outside of a greater authority was no truth at all, and Existentialism arose, causing men to doubt the meaning of, well, anything. This coincided with the rise of psychology in the mid-twentieth century, which displaced religion as the explanatory principle of man. Now, the self was not only central, it was worshipped. Existentialism gave way to nihilism in the mid-twentieth century. Nihilism led to societal revolt as seen in the Hippy movement and other threads. The Hippies grew up and became postmodernists, which meant that they no longer believed even in the reality of truth, as the modernists did. Thus, the rebellion was complete. Without any greater truth, man could live how he wanted. And thus came the spirit of anti-authoritarianism that we know so well today.