Learn Hardcore Korean: sueobi sijakdoegi jeone i munjereul pureo bwayo. - Try solving this problem before class starts.

수업 means class / lesson.
The particle -이/가 marks the subject of the clause, so 수업이 시작되기 전에 literally frames the time as before the class starts.
You could also say 수업 시작하기 전에 (more on that below), but 수업이 is natural when you treat the class as the thing that starts.

Why does it say 시작되기 instead of 시작하기? What’s the difference?

Both are common, but the nuance shifts:

시작하기 전에 = before (someone) starts (it) (more “active” wording)
시작되기 전에 = before it starts / before it begins (more “it happens” / event-focused)

In everyday Korean, 수업이 시작되기 전에 is a very natural, set-like way to say before class begins.

How does -기 전에 work grammatically?

Verb stem + -기 전에 means before doing X / before X happens.

In your sentence:

시작되다 (to begin/start)
→ 시작되- + 기 전에
= before it starts

You can attach -기 전에 to lots of verbs:

먹기 전에 = before eating
나가기 전에 = before going out
자기 전에 = before sleeping

Why is the first part placed at the beginning of the sentence?