На столе лежат три книги.

Because на столе expresses location: the books are already on the table.

After на when you mean where something is, Russian usually uses the prepositional case:

• стол → на столе

But на стол uses the accusative case and usually means movement onto the table:

• Книги падают на стол. = The books are falling onto the table.

So here:

• на столе = on the table, at rest

Russian uses на for many surfaces and places thought of as on something. A table is treated as a surface, so:

• на столе = on the table

If it were something like in a box, Russian would usually use в:

• в коробке = in the box

So the choice is based on the kind of spatial relationship:

• на = on a surface
• в = inside something

In Russian, simple present-tense sentences often do not use a separate word for are the way English does.

Also, Russian often prefers a position verb for physical objects:

• лежать = to lie, to be lying
• стоять = to stand
• висеть = to hang

So лежат does more than just mean are. It tells you the books are lying there, not standing upright or hanging.

That makes the sentence sound natural and concrete in Russian.

Because Russian often describes objects according to their physical position.

• лежать is used for things that are lying flat or horizontally
• стоять is used for things that are standing upright

Books on a table are often imagined as lying flat, so лежат is natural.

But if the books were upright, for example displayed vertically, Russian could use стоят:

• На полке стоят книги.

So the choice depends on how the speaker imagines the books.

Because the subject refers to three books, which is plural in meaning.

Even though numeral phrases in Russian have some special grammar, here the verb agrees with the idea of multiple items, so:

• три книги лежат

A singular verb like лежит would be wrong here in standard modern Russian.

After the numerals два, три, четыре, Russian nouns usually take a special form historically connected with the genitive singular.

So:

• одна книга
• две книги
• три книги
• четыре книги
• but пять книг

For книга, the form книги happens to look the same as the nominative plural, but here after три it is functioning as the form used after 2, 3, 4.

This is one of the most important Russian number patterns to learn.

In meaning, it is clearly plural: there are three books.

In form, after три, Russian uses the noun form associated with the genitive singular:

• книга → книги

This can be confusing because for this noun, that form is identical to the nominative plural:

• nominative plural: книги
• genitive singular: книги

So in this sentence, you should think:

• plural in meaning
• special numeral grammar in form

Yes. That is also correct.

Russian word order is flexible, and both versions are natural:

• На столе лежат три книги.
• Три книги лежат на столе.

The difference is mainly in focus or information structure:

• На столе лежат три книги.
  This starts with the location, as if setting the scene: On the table, there are three books.

• Три книги лежат на столе.
  This starts with the books themselves, so the books are more in focus.

So the original sentence is very natural when the speaker wants to introduce the location first.

Because Russian does not have articles like English a/an and the.

Russian leaves that information to context. So книги can mean:

• books
• the books
• sometimes some books

The listener understands the intended meaning from the situation, previous conversation, and word order.

Yes, but it sounds different.

• лежат is natural and physical: it describes the books as lying on the table
• находятся means something like are located or are situated, and sounds more formal, neutral, or less vivid

So:

• На столе лежат три книги. = natural, everyday, physical
• На столе находятся три книги. = grammatical, but more formal or stiff in ordinary speech

For everyday Russian, лежат is the better choice here.