I can't really speak much to international trends, but there are several factors to point to in the US. To begin, though, there should be some clarity regarding what we identify as a cult. It's a word that is loaded with all sorts of negative connotations, and therefore is avoided by religious studies scholars. They typically use the term "new religious movements" (NRM), which in fact comes from post-WW2 japan, where observers noted the rise of several strange new religions, a phenomenon that has often been attributed to the desire of many japanese to seek out new ways of understanding the world in order to help cope with the incredible trauma their country went through during the war. I bring up the concept of NRM in order to point out that this term doesn't typically get applied to Christian groups, even though many small churches come off as "cult-like" in their structures and membership behavior. This is important to note when discussing the supposed rise of cults, because back in the 19th century, before NRMs were common, there were certainly many cult-like churches, which means to some extent that the presence of cults didn't as much increase in the 20th c as it changed in terms of religious doctrines.

But as for the rise of NRMs, we see a pattern start appearing in the 1870s in the US and UK. In the UK, there was a sudden burst in Masonic-like groups focused around non-Christian and esoteric religious doctrines. My theory (which I've published on) is that with the increasing awareness of foreign, esoteric, and ancient religions due to books and travel technology, a man who made his living by selling things to Masons intentionally helped create and promote these groups to tap into this new growing market. Once these groups were formed, other people who had been interested in strange religions felt justified to start their own groups. In America, these were usually individuals who has been involved in the popular movements of spiritualism, mind healing, and unitarianism--movements that were generally accepting of a wide variety of religious views--and the main US innovators of the 1870s and 1880s were often aware of the UK Masons. By the late 1880s there were suddenly dozens of strange religion groups (NRMs), and they generally mixed all the above elements. As these groups grew, this emboldened white Americans in the 1890s to start forming groups for the so-called world religions, like Islam, buddhism, and hinduism (their American leaders had generally previously been in NRMs), and their success inspired even more NRMs to develop. By the start of the 20th century NRMs were popping up left and right. And To add to all the American-led groups, immigration brought several eastern religious leaders--who often represented non-mainstream sectarian factions or completely new religions. With the world wars and American racism causing great disillusionment with modern society, whites and blacks flooded these NRMs with the hopes of finding a better life and maybe helping to create a better world.

Surprisingly, according to some measures, despite the 60s and 70s being thought of as a period of even higher rates of NRM development, that might not actually be the case. What seems to be true though is that NRMs got a lot more attention from the press, alternative religions were more socially acceptable and often even seen as cool (particularly because of their association with drug use), and some groups got significantly higher membership numbers than what previous groups got. Part of the changes in the 60s and after was due to the emergence of the white baby boomers, who were wealthier, had more leisure time, and were more liberal than their parents. African Americans were often joining these groups either because they were associated with black empowerment or because they were seen as a solution to racism. Then, with the 1965 immigration reform, even more new foreign leaders came, further adding to the mix.

A lot of good overviews of 20th c NRMs have been written J. Gordon Melton and James R. Lewis. For 19th century stuff, see Godwin's Theosophical Enlightenment and my A History of Conversion to Islam in the United States vol 1.