It is a special use of the instrumental case with times. Compare вечером (in the evening), утром (in the morning), ночью (in the night), весной (in the spring), летом (in the summer), осенью (in the autumn).
You might think of it as an extension of the standard use of the instrumental case for the means (tool or instrument) by which something is done (писать ручкой - to write with a pen, using a pen, by pen).
The way to think about things can differ a little or a lot in different languages. The sense of зимой (using the winter, with the winter, by winter) is expressed in English as a slightly different notion: "in the winter" or "during the winter."
With regard to time in general, some languages view the future as ahead of us; some, behind us; some, above us; and some, to the west of us by the compass.
I have a question concerning the choice of the prefix "у-" for this sentence. It seems that that the sentence (certainly in the English translation provided here: "In winter, I leave home at nine") is intended to mean that the speaker regularly leaves home at nine (most likely in the morning), with the implication that the speaker returns home later on the same day. However, it's my understanding that the usual verb to express this idea of regular (daily) activity is "выходить": "Я выхожу из дома в девять." Unless I'm mistaken, "уходить" has the idea of leaving for an extended period of time, or even forever. In fact, I believe that Russian children threaten their parents to run away from home by saying "я ухожу из дома!" I would appreciate a reply from a native Russian speaker or an advanced student of Russian as to whether it's more appropriate to use "выходить из дома" rather than "уходить из дома" in a sentence implying regular (daily) activity.
To me, they sound about the same in this situation. Уходить, unlike выходить, can easily attach a destination (e.g., уходить на работу). So, to my ear, it does not imply leaving forever when combined with regular time intervals. It just implies there is a place you have to go to. On the other hand, just "Завтра я ухожу" does sound as if you were leaving for good.
Выходить can, at times, sound quite literal space-wise, like walking through the exit. If I am asking whether the person is going to school (work) or staying home, I can use "Ты сегодня уходишь?" The same with выходить, "Ты сегодня выходишь?" sounds way off. It might be OK if "выходить на работу" is what you would use with the job.
- выходить на работу is not about leaving at all; this expression means turning up to work.
Russian has several advebrs that mean a certain period of time:
- утром = in the morning
- вечером = in the evening
- днём = in the afternoon / during daytime (as opposed to hours after dark)
- вечером = in the evening/at (early) night
- ночью = at night (starts at around midnight)
- осенью = in autumn
- зимой = in winter
- весной = in summer
- летом = in spring
They are basically Instrumental forms of corresponding nouns. New adverbs are not made this way but these are very common.