In English we don't say "the winter" when referring to the season itself, it's simply "winter". EDIT: actually, maybe that's not true. "I ski in the winter" is just fine. I'm not sure exactly why, but I speak English natively and "the days of the winter" just sounds wrong to me.
I think the issue at hand is that in English, we favour the 's style possessive with definite nouns. The instances where we convert it to the of style possessive is when there's a modifier.
"That's the classroom of the teacher." (awkward)
"That's the classroom of the teacher who gives very low marks." (natural)
It works with seasons as well:
"The days of the winter were freezing." (awkward)
"The days of the winter of 1976 were freezing." (natural; could be phrased otherwise, but still still natural).
Do they mean that winter days (daylight hours) are short? (So, winter days are soon over; or winter days end quickly.)
Or do they mean, in a poetic sort of way, that winter will soon be over. (The days of winter end soon/are soon over - although I'm sure most people would say, "will end soon/ will soon be over".) Would an Italian-speaking person use a present tense to mean future in such a case?
This is not a correct English translation if this is the meaning of the phrase. In English you would say, in the winter the days are shorter or in the winter it gets dark earlier. The winter days end soon is the only English version that makes some sense. The winter days are soon over. It'll get warmer.