In the beginning, the world was much like Earth. But there exists an interstellar magical empire called the Eld. They came to the Earth and combined their DNA with a population of humans, creating Elves. Their purpose for doing so is that Elves would research and develop magic, slowly terraforming Earth into a magical ecosystem. When it's ripe enough (when reality has fully eroded), the Eld will return and plunder it, as they've done many times. Nothing remains when they're done.
The Elves don't necessarily realize they're a servitor race, but a lot of their cultural flaws are genetically programmed in. They're not actually in touch with nature, don't believe any Elf that claims otherwise. Many such beautiful, harmonious forests are actually dead, taxidermied graveyards, magically animated through necromancy to keep the flowers always beautiful, the animals always tame and obedient. Elves are all borderline sociopaths to a man, with Adventurer Elves being a rare exception infected with 'humanity', some recessive gene busting their genetic programming.
Dwarves are essentially Earth Elementals; they're not born or raised; they're carved out of the rock fully formed. This is why you don't see Dwarf women and children (though Dwarves are capable of interbreeding). They are the Earth's response to the introduction of magic, creating an antibody race that will fight it at every turn. The mining and work of the Dwarves is to make more of their kind, and to plunder gems and gold that are inherently magically charged. While they make many a magic item with these tools, the Dwarf crafter knows a Power Word killswitch that will Disjunction it. Dungeons are, unfortunately, magic turning this against Mother Earth. Underground constructions where dwarves are defeated, magic permeates, and Monsters are spontaneously generated in a perversion of Dwarven birth.
Orcs are a failed attempt at creating Elves, which were allowed to persist because the Eld found it amusing. Though ugly and savage and violent to other races, within their private homes, sheltered from the sky, orcs can be sweet, loving, tender, artistic. Always in hushed whispers, for their religion, their genetic program, teaches them that the Gods hate their existence, and make them suffer for a laugh, and punish them for all perceived goods they create or experience. Only by being a terror to the other races, they believe, will the gods laugh hard enough to relent their cruelty for another day. There is no good afterlife; all orcs suffer eternally before the gods, they're taught.
Gnomes and Goblins are the same race, both of them fey (more on fey later). Gnomes are vegetarians, but if they eat too much meat, they mutate into a Goblin, a savage and insane creature. Unfortunately, Goblins can breed true. However, a Goblin can be turned into a Gnome if turns down a meal of meat for as many times as the Gnome ate meat.
Halflings are the crossbreed between Humans and Gnomes.
Dragons are magic incarnate, their lifecycles a sign of the Eld's plans. They'll know their fruit is ready for harvest when the planet becomes an entirely Dragon-dominated ecosystem. Dragons created Kobolds out of their own blood. Every drop of dragon blood has the chance to grow into a kobold. Dragonborn are modified Dragon eggs, meant to be a servitor race to lead the kobold masses.
Fairies are born of the Earth, much like Dwarves, but are it's attempt to directly weaponize magic. They spread the infection of magic, but in the benefit of the natural world, providing antibody to Elven advancement. All the fey exist, but few are exactly as in the Monster Manual. Pixies and the like are born from the dreams of children, and so have a child's understanding of the world and morality. They die if the child awakens, so most fairies kidnap children and put them into diabetic comas in the Feywild. Changelings are homunculi, and always grow up into Elves. Elves lacking the Eld Programming, which breeds true...
Dryads are exceptionally beautiful, but are attracted to uglier creatures. CHA 3 is like CHA 30 to them. All dryad children are half-dryads, which become full dryads when they find their soulmate plant and bind with it. They take lovers for the only things a tree can't provide: Sex and conversation. You'll never win their heart, however.
Doppelgangers are a homunculus made up of melting ten or more people together, creating a shapeshifting being with a weak sense of identity. Besides their ESP and appearance-warping, doppelgangers can meld together and divide to switch and share memories and personality traits, which is necessary to prevent Alzheimer's in their plasticine brains. Loss of identity is what all Doppelgangers fear, but inevitable. Doppelganger blood can be imbibed to allow non-doppelgangers to meld for the duration of the drug. Doppelganger pregnancies are rare as there's always a chance of the embryo being permanently absorbed, but when they are born the child is invariably a Changeling, even if it's the child of two Doppelgangers.
Illithids, Aboleths, and other Aberrations are from the Far Realms either directly or through descent. The incursion of the Far Realms is the ultimate outcome of Magic, and if the fruit is overripe, the Eld won't be able to harvest it. A planet slipping entirely into the Far Realms is essentially letting a fruit rot to the worms.
The Gods and all Outsiders are the products of faith and belief. However, they are culture-specific. There are Elven Angels of Pride, Dwarven Angels of Greed, Orc Demons of Charity and Love, and so forth.
Elementals are direct manifestations of the Earth's will without any filter. Genies are what happens when an Elemental gains it's own ego and personality.
The Undead arise for all the usual reasons. Dead Werewolves rise as Vampires.
If an Aasimar and a Tiefling have a child, it's a human. If an Angel and a Demon (or Devil, or what have you) have a child, you get an Eld...