In the year 1885 famed French occultist Alexandre Saint-Yves was visited by a pair of mysterious men from the East, Rishi Bhagwan Das and the Indian prince Hardjij Scharipf. They told him of an ancient and secret land hidden at the center of the Earth, where the cultures of Atlantis and Manu lived on. Hardjij Scharipf taught Saint-Yves the secret Vattan alphabet from the dawn of time, still used in Agartha, and taught him many mystical secrets. These secrets included methods of communicating telepathically with the ascendant masters of Agartha. Amazingly, these masters strongly supported Saint-Yves' conservative-yet-radical political ideas of a highly structured class system leading to world harmony.
Hollow Earth theories weren't anything new at this point, having been a part of mythology since time immemorial, and more recently having been put forth in some pseudo-scientific detail by Edmond Halley in 1692. Saint-Yves managed to make Agartha stand out through carefully tying it in to his views on politics, religion (the secrets of Agartha will be revealed when Christians live up to the Commandments given by Moses), and a good bit of story-telling. His accounts of Agartha describe it as a very real and current place, accessed through the Tibetan Himalayas.
The story was reinforced when Antoni Ferdynand Ossendowski, a well-known travel writer, wrote in his 1922 book Beasts, Men and Gods about a Buddhist myth concerning a subterranean kingdom known as Agharti. While there aren't many followers of Saint-Yves around today, the myth of Agartha continues to float around among those interested in such things, and theosophists have adapted Agarthi as a large cave system underlying Tibet that is inhabited by evil spirits known as asuras.
As you may have noted, Agartha has a number of names/spellings, including Agartta, Agharti, Agarta, and Agarttha. Agartha is the most common, but the others live on and new ones are occasionally created, as in the city of Agart in Final Fantasy IV, which is the portal to subterranean realms.