(Yes, I know George R. R. Martin has explicitly stated that there is no scientific explanation for the unpredictable seasons in the saga--that it's magic. But I came up with this explanation just for fun.) The world of A Song of Ice and Fire is a parallel universe to ours. In that parallel universe, around 14,000 years ago, a comet impacted with the Moon. Because of that, large chunks of the Moon bombarded the Earth in a meteor shower. That meteor shower drastically altered the geography of the Earth. The comet impact left the Moon dramatically smaller. A gravitational effect of the smaller Moon is that the Earth's rotational axis has been permanently destabilized. That's why the planet has inconsistent and unpredictable seasonal lengths. The altered planetary geography (which includes extremely large oceans, an abundance of gigantic mountains, exceptionally strong ocean currents and extremely powerful prevailing winds) also contributes to these unpredictable weather trends. Along with the altered landscape, another effect the meteor shower had on the Earth was a brief but considerable disruption of the atmosphere's blocking of cosmic radiation. During that disruption, the radiation that hit the Earth caused mutations across life on Earth. That's how animals such as shadowcats, dragons and krakens and plants such as weirwood trees came into being. Also, direwolves developed physical mutations that enabled them to avoid extinction. Humans were also effected. That's how human-derived species such as the Children of the Forest and the White Walkers came into being. It's also the reason the Targaryens have platinum hair and violet eyes and the Lannisters are always blond. Another mutation the radiation caused in humanity was the capacity to mentally summon extra-dimensional forces such as energies and entities from hyper-dimensional pocket universes and then use those extra-dimensional forces to manipulate the fundamental interactions of nature and even aspects of the space-time continuum itself. That's what "MAGIC" is. Due to a macroscopic form of quantum entanglement, universes in parallel with each other converge towards specific configurations despite their divergent timelines. As a result, despite vast historical differences, certain circumstances are paralleled in both our universe and the A Song of Ice and Fire universe. Examples include the fact that the culture of modern Westeros is like the culture of medieval Europe and the Common Tongue (the dominant language of Westeros) is identical to modern English.
Direwolves are real-life animals that went extinct about 10,000 years ago. No, "mutation" needed. I also seem to recall other Ice-Age Megafauna still being alive in Westeros like Mammoths, though I may be mistaken.
I don't know too much about the science, but magic aside, can the planet where Westeros exists just orbit its sun more slowly thus allowing for the extended winters? Now I know that the seasons are supposed to be irregular, but if they just happen more slowly cannot their springs and falls just vary in a way similar to ours?
Supervolcano in Valyria determines the length of the seasons based on the amount of ash it releases into the atmosphere. At least, that's always been my explanation for those irregular seasons. Because from the way they're described, Westeros' winters do sound like nuclear winters. Only problem with that theory is that seasons probably have been irregular even before the Doom.
They've been irregular for a long time, probably at least a thousand years, it is hard to say how accurate their histories are. Real-life Direwolves weren't especially large. Shadowcats = Some variant of tigers, jaguars or leopards Lizard-lions = Alligators Krakens = Giant Squid Targaryens aren't the only ones to have Valyrian features like violet eyes and silver-blond hair. It is just that most of their people died when their island nation when kaboom and took out most of the surrounding area. Not all Lannisters are blond. Their genes are actually suggested to be recessive, but George has also admitted that genetics is not a strong suit of his and the books are not scientifically accurate. Radiation kills stuff. As much as comic books make it sound great, DNA rarely benefits from irradiation.
Surviving a fire unharmed, giving birth to anthropomorphic black clouds that assassinate people, and reviving the dead are all pretty much magic -- unless you invoke advanced nanotechnology perhaps. The bizarre flora and fauna I'll attribute to genetic engineering. The Valyrians were purported to have had advanced technology now lost to the world. The opening graphics of the TV series make the world appear to be an orbital or a segment of a shell-type Dyson sphere, and the sun to be artificial. Anyway, we've seen the "sun" rising and setting so that makes that theory problematic. I'll discount that. The characters talk about Winters lasting years, so a very elliptical orbit wouldn't fit the bill. The simplest explanation would be for the sun to be an irregular variable star.