The popularity of bootlegs was due to the novelty of hearing music not otherwise available.

There were really two categories of bootleg recording. One kind were informal copies of cassette tapes copied and traded among fans. Quality varied widely, especially due to additional tape hiss added on each generation. These were very inexpensive, sometimes just the cost of the blank cassette.

The other kind were commercial bootlegs, where a dodgy offshore company would press unauthorized LPs. Often these were more expensive than the official release due scarcity and lack of an above-board distribution network.

Bootlegs can be interesting from an historical perspective. Live shows often contain songs that are not in their final form, and give insight into the creative process. The Grateful Dead was the most famous for having bootleg copies available for nearly concert, and you can look back at how the set list has changed as the years went by.

There is one example from Pink Floyd bootlegs where they show when the band changed a few words in the lyrics to the song "Echoes" off of the Meddle album. They preformed the original set of lyrics for a dozen shows, and then switched to the new version. Then about three shows later they messed up. Dave sang the old lyrics and Rick sang the new lyrics.

Frank Zappa pulled a clever trick where he acquired bootleg copies of his live shows, and then released them on his own record label in the Beat the Boots series. This gave him the income but also dropped the market out of the illegal copies.

(note, the answer here reflect US experience; although the US is one of the largest producers and consumers of recorded music, its experience does not reflect other countries. If anyone wants to chime in on Europe, Latin America, Africa, Asia, I encourage you to do so)

The apparent mushrooming of bootlegs in the 1960s had much to do with both the expansion of the consumption of music and the proliferation of recording technology. These two factors worked together in a synergistic fashion. As the consumer audience for music broadened after the Baby Boom, so to did the appetite for new musical experiences. The advent of reel-to-reel magnetic recorders allowed individuals to record radio broadcasts or even live concerts. Furthermore, a specialized music press helped to create a climate which emphasized there was more to enjoying a musical experience than just the commercially available records or concerts.

It's important to realize that this musical culture was not without precedent in the US. American copyright laws as they pertained to musical recordings was notoriously vague until 1971. Most federal copyright law held that companies held the rights to the medium (vinyl record, magnetic tape, 8track, etc.) but not the music itself. What consumers did with the music after buying the recording was considered by some jurisprudence to be artistic modification of their property. The US recording industry tried consistently to expand the definition of federal copyright, but bootlegs operated in a gray zone until the 1971 Sound Recording Act. For example, the creators of the first big bootleg "hit," Bob Dylan's Great White Wonder tried to argue since they never used Dylan's name nor likeness, they were not violating any law (for all you know it was a damn good impersonation). Jazz, opera, and blues aficionados of the 1910s and 20s had created a lively and vibrant market of special imprints, live recordings and other bootlegs. The postwar boom in music and consumerism far exceeded even the heady days of the 1920s and bootlegs proliferated accordingly.

Some of these bootlegs were relatively cheap with poor audio quality, but other recordings reflected the increasingly sophisticated musical palette of music consumers. The Smokin' Pig and Wizardo labels had their own unique covers and mixes. These labels, often operating out of existing musical shops, presented themselves as alternatives to the more consumerist mainstream recording industry. Of course, artists themselves were also a part of this subculture of music. One urban legend around an early release of "What's the New Mary Jane," came when Lennon, an avid collector of bootlegs, exchanged an early copy of the song in exchange for a number of rare recordings. Artists were not the only source for unused masters or demos. The security at recording industry's vaults was notoriously lax. The theft or copying of these materials became irresistible when famous bands' projects seemed stuck in limbo. The recording sessions for Let it Be and Get it Back were widely bootlegged during these albums' gestation.

Although bootlegs never went away, their numbers did decline in the 1970s and 80s, especially for mainstream music (outliers like punk, hiphop, and rap were notable exceptions). Partly, this was because of the further growth of private home recording allowed consumers to make their own private mixes and recordings. Also, not only did studios tighten their security, but also became more amenable to releasing live and other unique recordings under special imprints and labels. Finally, musical tastes changed, often into formats like disco that required more sophisticated sound mixes than the music of the late 60s, so the variable audio quality of bootlegs became more apparent. However, to reiterate, bootlegs never really went away in music, but became a subgenre in their own right. Although the 1960/70s was a unique time for bootlegs, they were neither the first nor were they the last examples of this dimension of musical consumption.

Sources

Cummings, Alex S. "From Monopoly to Intellectual Property: Music Piracy and the Remaking of American Copyright, 1909–1971." The Journal of American History 97, no. 3 (2010): 659-681.

Heylin, Clinton. Bootleg: The Secret History of the Other Recording Industry. New York: St. Martin's Press, 1994.