Learn Russian: Мы кормим птиц зимой.

Птиц is in the accusative case because it is the direct object being fed. Because birds are alive (animate), their plural accusative form matches their plural genitive form. To form the plural genitive of the feminine noun птица, we drop the -а ending, leaving just птиц.

Why is the word for winter (зима) in the instrumental case (зимой)?

In Russian, to say that something happens in a certain season or during a part of the day, you use the instrumental case. So зима (winter) becomes зимой (in the winter), just like утро (morning) becomes утром (in the morning).

The English translation says "in winter", so why is there no preposition like в before зимой?

When using the instrumental case to express time for seasons or parts of the day, Russian does not use a preposition. The instrumental noun does the job all on its own. Adding в here would be grammatically incorrect.

What is the infinitive of кормим and is it a regular verb?

The infinitive is кормить (to feed). It follows standard second conjugation rules, which is why it takes the -им ending for мы. Just watch out for the я (I) form: it adds an extra consonant to become я кормлю.