(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.