Learn Farsi: دی ماهِ زمستان است.

That is the ezāfe (the connecting "-e" sound). It links the noun māh (month) to zemestān (winter). Even though English says "winter month," Farsi literally says "month of winter," using the ezāfe to connect the two nouns.

The translation says "a winter month," but I don't see a word for "a". Why is that?

In Farsi, when you are stating a general fact or putting something into a broad category (like saying Dey belongs to the group of winter months), you usually don't need an indefinite article like یک (yek) or the -ی (-i) suffix. Stating ماهِ زمستان (māh-e zemestān) is perfectly natural on its own.

What exactly is دی (Dey)?

دی (Dey) is the tenth month of the solar Iranian calendar. It corresponds to late December and most of January, making it the very first month of winter (zemestān) in Iran.

Why does the sentence use the full word است (ast) instead of the spoken ending?

است (ast) is the formal, written way to say "is". This sentence is written in a standard literary style. In everyday spoken Farsi, people would shorten this and attach an -e sound to the end, saying دی ماهِ زمستانه (Dey māh-e zemestāne).