Obviously, low fidelity refers to sonic quality, as in a low fidelity recording or a low fidelity tape. It is naturally trebly, neither reaching crispness on the cymbals, or timbre in the bass. It is often the sign of a cheaply recorded artefact, or a cheaply constructed playback unit. A bootleg album is typically low-fidelity. A home-made tape usually is.
To appreciate what lo-fi meant in its original, classic sense, think of the Kingsmen's "Louie, Louie"; so murky the lyrics were believed to be obscene (they weren't). Think of the rarest 45's from the puniest 60's garage bands, how they're all fuzz. Catch a whiff of overheated vacuum tubes and tinny transistors.
In the 1970's, punk music was a low-budget affair, and most of the seminal punk records of the time were recorded on cheap, second-rate equipment. That same tinniness from the cheaply made 60's recordings is evident on the punk records, too. In some cases, they may very well have been recorded on the same old equipment. The trebley static and hiss of these records is as much a part of their sound as the actual notes played.
When the compact disc appeared, there was a small but significant faction of music consumer who resisted them; to their ears, the crystalline sound of remastered albums brought out a fullness of sound that sounded alien to them; the DNA of the song itself was seemingly altered. There were also those musicians who grew up listening to hissy records, and had spent their lives dreaming of making hissy records of their own someday.
By the 1980's, professional recording equipment had become small enough and affordable enough that small-time producers would often have a little 4-track and soundroom available at home for recording demos and even albums. Although better equipment had become available at professional studios, including digital technologies, few of these bands had the resources to pay for extended studio time; thus, their 4-track home recordings often had that same lo-fi quality as many of their heroes' albums did. Some of these artists, rather than trying to mask the low fidelity of the recording, would make use of the sonic limitations; experimenting with white noise, free-form, art-noise, to incorporating pure pop and melodic rock into the lo-fi environment.