The Talmud (Rosh HaShanah 7a) uses this verse to suggest we usually count the years from Rosh HaShanah every Tishrei, but we count the individual months from Nissan. The names of the months we use are not even Jewish in origin (Yerushalmi, Rosh HaShanah 1:2), so why do we use them? By mentioning the names, Ramban writes (on Shemos 12:2) we remind ourselves that we are in exile, still awaiting Mashiach, speedily in our day.
The Talmud (Megillah 13a) says that Achashverosh and Esther met in this month because Teves is a naturally colder month, and the warmth of bodies would be more enjoyable. This way, Achashverosh would be naturally predisposed to like Esther more than the other women. In this seemingly “natural” manner, H-Shem works His miracles.