Narnia had been under the grip of Her Imperial Majesty Jadis, also known as the White Witch. In the absence of Aslan (which symbolizes good, and perhaps heat?), the Chatelaine of Cair Paravel took over and covered all of Narnia in a blanket of cold. Knowing what we know about cold being the absence of heat, it was more likely she sucked the very energy out of the land and its peoples.
Beings from "our world" often entered Narnia through chinks and chasms in the rocks, or through other means, like the wardrobe the Penvesie children found accidentally. However, they could not enter from the skies, for it was a dome impenetrable by mortals.
The flying horse, Fledge, was actually the cab-pulling earth horse, formerly called Strawberry, who was chosen to become a talking creature and who also acquired wings. If you look carefully, Strawberry / Fledge is flying over the chilly landscape, hiding and peeking amongst the clouds.
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