Chcę wypożyczyć książkę.

Because Polish often drops subject pronouns when they are clear from the verb form.

Chcę already means I want, so ja is not necessary.
You could say Ja chcę wypożyczyć książkę, but that would usually add emphasis, as in I want to borrow the book.

Chcę is the 1st person singular present tense form of the verb chcieć (to want).

So:

• chcieć = to want
• chcę = I want

This verb is a bit irregular, so the form does not look exactly like the dictionary form.

Because after chcę Polish usually works like English want to + verb.

So:

• chcę czytać = I want to read
• chcę kupić = I want to buy
• chcę wypożyczyć = I want to borrow/check out

The second verb stays in the infinitive.

Because książkę is in the accusative case, which is used here for the direct object of the verb.

The basic form is:

• książka = book

But in this sentence, the book is the thing being borrowed, so Polish changes it to:

• książkę

For many feminine nouns ending in -a, the singular accusative ends in -ę.

In this sentence, wypożyczyć książkę most naturally means to borrow/check out a book, especially from a library.

A useful nuance:

• wypożyczyć often suggests borrowing from an institution, such as a library, or renting something temporarily
• with książkę, the usual meaning is borrow/check out a book

So this is a very natural verb choice in a library context.

This is a very common question, because pożyczyć can be tricky.

• wypożyczyć = to borrow/check out, often from a library, shop, or institution
• pożyczyć = can mean lend or borrow, depending on the structure

Examples:

• Pożyczę ci książkę. = I will lend you a book.
• Pożyczę od niego książkę. = I will borrow a book from him.

So wypożyczyć książkę is clearer in a library situation.

It is perfective.

That means it presents the action as a single completed whole: borrowing/checking out the book.

Its imperfective partner is wypożyczać.

Compare:

• Chcę wypożyczyć książkę. = I want to borrow/check out a book.
  (one complete action)
• Chcę wypożyczać książki. = I want to borrow books regularly / be borrowing books.
  (repeated or ongoing activity)

In your sentence, the perfective verb sounds very natural.

Yes. Polish word order is fairly flexible, because case endings show what each noun is doing.

The most neutral order is:

• Chcę wypożyczyć książkę.

But you may also hear:

• Książkę chcę wypożyczyć.
  This puts more emphasis on the book.

Changing the order changes focus or style, not the basic meaning.

This word is hard for English speakers because of the consonant cluster.

A rough guide:

• ch sounds like the ch in Scottish loch or German Bach
• then comes c, pronounced like ts
• ę is a nasal vowel, but at the end of a word it is often pronounced only weakly nasally in everyday speech

So chcę is roughly like kh-tseh, though the real Polish sound is tighter and more compact.

These are Polish nasal vowels.

In this sentence you have:

• ę in chcę
• ą and ę in książkę

A few practical points:

• ę at the end of a word, as in chcę, is often pronounced less strongly than learners expect
• ą in książkę is nasal too, but its exact sound depends somewhat on the following consonant
• you do not need to exaggerate the nasal quality

For a learner, it is better to say them lightly and clearly than to overdo them.

It is completely grammatical and natural, but in a real conversation, especially with staff, Polish often prefers a softer or more polite form.

For example:

• Chciałbym wypożyczyć książkę. = I would like to borrow a book.
  (said by a man)
• Chciałabym wypożyczyć książkę. = I would like to borrow a book.
  (said by a woman)
• Czy mogę wypożyczyć tę książkę? = Can I borrow this book?

So Chcę wypożyczyć książkę is correct, but it can sound a little more direct than English I’d like to borrow a book.