Learn Russian: Студентам не сидится в аудитории: не верится, что скоро каникулы! - The students can't sit still in the classroom: they can't believe that vacation is soon!

This uses the involuntary state structure you are learning. By putting the subject in the Dative case (Студентам) and adding the reflexive ending -ся to the verb (сидится), it shows they have no control over the action. They aren't just choosing not to sit; they physically can't sit still because they are too excited.

In не верится, who is the one that can't believe it? Why is there no pronoun?

The students are still the ones who can't believe it. Because the experiencer (студентам) was just established in the first half of the sentence, it is dropped before не верится to avoid sounding repetitive.

Why use в аудитории instead of в классе for "in the classroom"?

Аудитория is the specific word used for university or college classrooms and lecture halls. Because the sentence is about студенты (university students, not school children), аудитория is the correct word here.

Why is каникулы used here for "vacation", and why does it look plural?

Каникулы specifically means school or university holidays (adults taking time off work would use отпуск). Also, каникулы is a plural-only word in Russian, just like "scissors" or "pants" in English, even when you are only talking about one upcoming vacation period.