Mystically: your fictional people remember their past incarnations. With a bit more of handwaving this ceases being mystical and can actually go scientific.
Mystically, version 2: your beings can access the Akashic Records. Depending on the mystical school describing these, they may contain from just thoughts and desires of everyone who has ever lived to each and every detail of the whole history, past and future, of each particle in the universe, so taylor it to your needs.
Genetically: as seen in Assassin's Creed. This has an advantage that you can limit the amount of information that a species can carry from one generation to the next, if you wish.
Genetically, version 2: as seen in Steven Universe. Gems are born when a planet's crust is seeded with a kind of genetic matrix for their species. The matrix contains not only the instructions for their shape and composition, but also all the knowledge that they will have upon birth.
Genetically, through horizontal transfer: if genes can carry memories, these memories can be implanted into someone else through viral infection. Virii will sometimes incorporate genes of one host, and when they infect new hosts later on, the more recent host will have some of the previous host's genes implanted on their infected cells. This is called horizontal gene transfer and it is a thing in nature. Alternatively this may be achieve through genetic engineering/therapy, for a more controlled and accurate effect.
Telepathically: as seen with the Zerg species in Starcraft. A Zerg learns all they need to know via telepathy, being instructed by the hierarchy that spawned them. They are guided, in ascending order of rank, by hatcheries, overlords, cerebrates and on top of that an overmind.
Technologically: you can have a brain implant installed on the brain of a newborn to kickstart its brain with memories from other people (or even fabricated memories). See Ghots in the Shell (movie and animé), and the awesome sci-fi books Old Man's War and its sequel The Ghost Brigades by John Scalzi. Though your species may have no technology of their own, a more advanced benefactor may implant stuff on them.
Technologically, version 2: Again an implant, but this time to link everybody together like a hive mind. The technological version of telepathy. Star Trek's The Borg work like this. This way everybody can access the same knowledge, or maybe a secured version per family. This technique could also allow a human version of cloud computing. See Joe Haldeman's The Forever War and Forever Free for a variation on this theme.
Magically: again as seen on Assassin's Creed. At some point the protagonist of the original game transfers his memories to a set of discs (such discs are technological in nature, but that technology is advanced enough that it is indistinguishable from magic for all purposes). One of his descendants absorbs those memories from the discs a few centuries later. This memory transfer method does not rely on the genetic memory transfer method on which the series is based.
Magically, version 2: as seen in Steven Universe. Gems can fuse among themselves to form new beings. These beings have the memories from the gems that fused. They are also able to use any component's wisdom, insight and personality to judge, handle, process and combine any other component's memories, thus being able to have much more knowledge than the mere sum of each isolated component's memories.
Organically: those trees from James Cameron's Avatar. People can connect to those to upload and download memories. Seems like whole forests are interconnected as well.
Failing all those... Maybe the species reproduces by binary fission, like bacteria. Each child will have the memories of their parent. I have read that this is the case with starfish when an arm regenerates a whole body, but I can't find any sources confirming this now.