Learn Hardcore Russian: Зимой мне всё чаще не спится, поэтому я читаю книгу. - In winter I increasingly can’t sleep, so I read a book.

Зимой is the instrumental case of зима (winter) used to express time/season when something happens.
Common pattern: [season in instrumental] = in (that season).
Examples: летом (in summer), весной (in spring), осенью (in autumn), зимой (in winter).
You can also say зимой without a preposition; it’s a set, very common time expression.

What is the grammar behind мне не спится? Why not just я не сплю?

мне не спится is an impersonal construction with the reflexive verb спаться. It describes how sleep is (not) happening to someone, often meaning you can’t fall asleep / can’t sleep (even if you want to).

я не сплю = I’m not sleeping (a neutral fact; maybe I’m awake on purpose).
мне не спится = I can’t sleep / I’m unable to sleep (often involuntary, a state).

Why is мне in the dative case?