Turn on any broadcast or crack open any nonfiction book in the Federation, and Earth will probably be mentioned at least once. Although in the case of broadcasts, chances are there will be at least one show dedicated to the far-flung lonely planet of the Milky Way Galaxy.
To Earthlings, though, it's a home planet in a vast ocean of stars. Most Earthlings have no concept of the Federation looming beyond other galaxies, nor do they care. To be an Earthling, as those in the Federation quip, is to be blissfully ignorant.
It's important to note that primates existed naturally on Earth long before the first Federation settlers arrived thousands of years ago. Due to a lack of record keeping (and care) by the earliest Federation governments to intervene with Earth's natural course, it is not known exactly how modern humans are related to the ancient ancestors to be deposited in the Cradle of Civilization. Modern scholars, both on Earth and in the Federation, believe that the humans brought to Earth mostly kept to themselves for generations, possibly splitting into separate tribes as their numbers grew. All that is known for sure is that these Federation humans are linked to ancient Mesopotamian civilizations such as Sumer and Akkad. It is unknown if they were at all related to rising civilizations on other parts of Earth, such as Indo-America and Aboriginal Australia.
As for what brought these humans to Earth, only the Federation governments know for sure. Modern Earthlings are kept completely in the dark about their extra-terrestrial origins, and their distant cousins throughout the universe are taught that Earth was originally a penal colony for the thousand or so exiles who committed a crime grievous enough to incur enough wrath for the Federation to go through the trouble, but not enough for the death penalty. The biggest rumors include the Temple being somehow involved, and some Yahzen scholars postulate that such humans created enough of a schism to be exiled for the sake of Temple control.
It is important to note that at the time of the great exile, Federation technology was just potent enough to make it happen - but that it was a great undertaking incomparable to what the Federation is capable of today. The humans left on Earth would have been given nothing but the clothing on their backs and whatever food was still in their stomachs.
Over time, Earth was not forgotten. Federation civilians acquired a curious interest in Earth, particularly how their cousins fared on their own. As generations passed, this curiosity transformed into entertainment. While Earthlings forgot their extraterrestrial origins, whether by design or accident, they developed their own civilizations and technologies independent of the Federation. This allowed Federation government and media to keep a close eye on Earthlings, even going as far as recruiting "plants" who would join Earth's various civilizations and report back on developments. Over centuries, this included broadcasts and journalistic documentation that utilized tech not yet found on Earth. The construction of the Great Pyramids was broadcasted live to the Federation, while copies of the great epic poems and novels of their times were translated and disseminated throughout the universe, unbeknownst to their authors.
Interest in Earth only grew as civilizations became more complex and distinct cultures grew across the planet. Earth is often used as an underdog story in Federation literature, while also providing convenient scapegoats to naughty children and their flummoxed mothers. (The phrase "I'll send you to Earth" is commonly heard while children are punished.) Academics who focus on Earth are not unheard of, and many careers have been built on translating, reporting, and teaching about one of the universe's biggest petri dishes for culture and the study of the human mind.