Learn Hardcore Korean: doseogwan chaegeun jikjeop jettae bannaphaeya hae. - You must return library books yourself on time.

Breakdown of doseogwan chaegeun jikjeop jettae bannaphaeya hae.

What does the particle 은 in 도서관 책은 do here? Why not 을?

은/는 marks the topic: it sets up “as for library books…” and often sounds like a general rule or contrastive statement.
Using 을/를 would mark it simply as the object with no topical nuance: 도서관 책을 직접 제때 반납해야 해.
Both are grammatical. 은 feels like you’re stating a policy/guideline (“When it comes to library books…”), while 을 is a straightforward statement about what must be done.

Who is the implied subject? There’s no “I/you/we” in the sentence.

Korean often drops pronouns. The subject is recovered from context:

In conversation, 해야 해 commonly targets the listener, so it often means “you must…”
It can also mean “I/we must…” if that’s clear from context.
On signs/notices, you’d usually see the more formal 해야 합니다 addressing the general public.

What exactly does 직접 mean? Is it the same as 스스로 or 혼자서?

직접: in person, without a proxy or intermediary. Emphasizes doing it yourself rather than having someone/something else do it.
스스로: by oneself, of one’s own accord/effort (self-driven). Not about intermediaries, but about self-reliance.
혼자서: alone, without company or help (physically by yourself). Examples:
우편이나 대리인 말고 직접 반납해. (Not by mail or another person.)
숙제는 스스로 해야 해. (Self-driven effort.)
혼자서 도서관에 갔다. (Went alone.)