Learn Hardcore Korean: chaekjang wieneun mulgeoni manhi isseoyo.

위에는 breaks down as 위 (“top/above”) + 에 (location particle for existence) + 는 (topic marker). Together it means “as for on top of (the bookshelf).” You’re making the location itself the topic of the sentence.

Why is 물건이 marked with -이 instead of the topic marker -은?

Here, 물건 (“things”) is the grammatical subject of the descriptive verb phrase 많이 있어요. The adjective 많다 (“to be many”) takes a subject marked by 이/가, not a topic marker. If you used 물건은, you’d be topicalizing “things,” which changes the nuance.

Why is 많이 used instead of 많다?

You’re saying “there are many things” by using the adverb 많이 (“many/much”) to modify 있어요 (“exist/there is”). 많다 is an adjective (“to be many”); its adverbial form 많이 describes how they exist.

What does 있어요 mean, and why isn’t it a copula?

있다/없다 are verbs indicating existence or possession, not copulas. 있어요 is the polite present form of 있다, meaning “there is/there are.” A copula (이다) links a subject to a noun/adjective, but 있다 expresses that something exists somewhere.

Can I drop the topic marker 는 in 위에는 and just say 책장 위에 물건이 많이 있어요?

Yes. Omitting 는 gives you a more neutral statement without topical emphasis: 책장 위에 물건이 많이 있어요. The sentence is still correct and means the same thing.

Why can’t I use 에서 instead of 에 for the location?

에서 marks the location of an action (e.g., 공부해요 – “study at”). 있다/없다 describe static existence, so they always take 에 to mark where something is or isn’t.

How would I say “There aren’t many things on the bookshelf”?

You can negate 많다 with -지 않다:
책장 위에는 물건이 많지 않아요.
Or you can use 별로 없다 (not many/not much):
책장 위에는 물건이 별로 없어요.

Can I change the word order without changing the meaning?

Yes. Korean word order is flexible. For example:
물건이 책장 위에 많이 있어요.
This still means “There are many things on the bookshelf,” but it shifts a bit of emphasis onto 물건이.