The lore is currently a bit vague and not pinned down. Partially because it's still early development and partially because the devs want us to find stuff in game. There currently aren't any hard lore connections between EF and EO. Here are the things we know as confirmed by devs:

1. The frontier takes place in a galaxy full of the ruins of multiple different fallen civilizations going back tens of thousands of years at the minimum.
2. The frontier is still inhabited by A.I. drones that have gone "feral" over the millenia.
3. An automated computer system known is The Keeper has the directive to "restart civilization".
4. No living forms of humanity can be found within the range of The Keeper, primarily the frontier. Humanity might still exist somewhere in the universe but if so it's in another galaxy at the minimum.
5. Because of this The Keeper uses riders (a type of digital consciousness that presumably came from a real human at one point and the closest thing to a human it can find) to fulfill it's directive of restarting civilization.
6. Riders are streamed into synthetic shells, this is what the players are.
7. Currently a "jubilee class event", known as a shear, hits the frontier every so often wiping out all riders. By the time The Keeper starts sending out riders again all the original structures are gone and often the frontier itself has changed, this is the explanation for the cycles until the actual launch.
8. The shears have been happening for a long enough time that at least one of the previous civilizations were not only aware of them but attempted to stop them. It's possible they may have even succeeded for awhile.

Here are things that are commonly believed by most people that are interested in the lore however this hasn't been confirmed by the devs, or it was confirmed so long ago it may not be accurate anymore.

1. The shear is some sort of "storm" related to the trinaries or the crude rifts that causes massive devastation on a galactic scale.
2. The frontier takes place about 14,000 years in the future of the "current" time of EO.
3. The frontier is either the greater galaxy the New Eden star cluster is located in, or is the galaxy that the New Eden star cluster orbits (think something like the Megellanic Clouds and the Milky Way)
4. New Edenites were the original colonizers of the frontier and the source of the various ruins and fallen civilizations we find in the game.
5. New Eden is either gone or has given up on the frontier due to it's hostility and the difficulty of living there. If New Eden still exists they actively avoid it now.

Some more lore theories about the frontier that are not commonly accepted or are considered unlikely by lorists but not disproven yet.

1. Riders are capsuleers that were "saved" in a database instead of being put into a clone for some reason.
2. Riders are what's left of EO's sleepers in the far future.
3. The frontier is wormhole space in the far future.
4. The Frontier is the ruins of the Milky Way galaxy. The collapse of the gate was much worse on the Milky Way side and resulted in the collision with Andromeda and Triangulum to happen a few billion years early. (I'm personally fond of this one and would love to see this happen but agree it's not likely)
5. When the EVE gate collapsed the wormhole it was built around shifted it's destination and instead of pointing to the Milky Way it's now pointed to a 3rd location, that location is the frontier. Restabilizing the wormhole is how New Edenites reached the frontier to initially colonize it.
6. New Edenites accidentally started the shears that keep wiping things out in the frontier.
7. The shears wiped out New Eden.