Learn Russian: Зимой мы часто ездили по лесу. - In winter we often drove through the forest.

The preposition по requires the Dative case. The word лес is masculine, so it takes the Dative ending -у, becoming лесу. In this context, по means moving along, around, or through a space.

Why do we use the verb ездили instead of ехали?

Ездили is a multidirectional verb of motion, used for habitual trips or moving in various directions. We use it here for two reasons: the action happened часто (often), and driving по лесу implies moving around the forest in different directions rather than making one straight trip to a specific destination.

If зима means "winter", why is it Зимой here? Why isn't there a preposition like "in"?

To say "in" a certain season, Russian uses the Instrumental case without any preposition. The feminine noun зима takes the Instrumental ending -ой to become зимой (in winter).

The translation says "through the forest". Could I use через instead of по?

You could say через лес, but it changes the meaning slightly. Через means crossing from one side directly to the other. По лесу means driving around within the area of the forest. English often uses "through" for both, but Russian makes a clear distinction!