Aside from the Red Gyarados, which is Magikarp forced to evolve by Team Rocket that retained its red coloring (if I'm reading this correctly), the anime doesn't really expand on the origins of shiny Pokemon much. If I had to guess, I'd say a lot of shinies are just born/made regularly, just with a different coloration, maybe as a result of a rare genetic mutation.
Interestingly, some shinies in the anime have the ability to do something that others of their species can't, like the shiny Shuckle in Johto can make a special medicine with its juice, or the shiny Genesect in the Genesect movie is much faster than the normal, other four. However, as some shinies are leaders of a pack or such, the most likely reason for why some anime shinies are shiny is to visually distinguish the Pokemon as to say, "hey, this one is special," "this trainer is special/has some significance," or, "this Pokemon is different from when it appeared earlier in the wild or with another trainer." If it's not to visually distinguish from another, I imagine the show makers just throw in random shinies from time to time because, uh, why not? Shinies are cool.
But yeah, it's likely just a genetic mutation or something during a normal birth. I don't think anime shinies are born on a perilous path after their mother struggled to carry them away from persecution and lightning lifts the egg of the shiny in the air while wind wildly whips around the egg, sending the mother off the cliff and making her die, cracking the egg as it hits the ground, forcing the shiny to struggle as a limping, homeless orphan, getting bullied and beaten until they find the right trainer or something, no, that's unrealistic. In fact, I don't think we've actually seen a shiny born in the anime, although I maybe might be wrong.
