Learn Italian: Questi sono i libri nuovi.

Questi is the masculine plural form of questo (this). Because it refers to i libri (the books), which is a masculine plural noun, we must use Questi. If we were talking about a feminine plural noun, like 'le scarpe' (the shoes), we would use Queste.

Why isn't Questi placed right next to libri?

In this sentence, Questi acts as a pronoun, meaning it stands on its own to mean 'These'. If you wanted to say 'These books are new', you would use it as an adjective right next to the noun: Questi libri sono nuovi.

Why is nuovi at the end of the sentence?

In Italian, descriptive adjectives usually come directly after the noun they describe. So instead of 'the new books', the phrase is built as 'the books new' (i libri nuovi). Notice that nuovi ends in -i to match the masculine plural noun libri.

Could we leave out the i and just say Questi sono libri nuovi?

Yes, but the meaning changes slightly! Using the definite article i makes it specific: 'These are the new books' (perhaps the ones we were waiting for). If you drop the i, it translates to 'These are new books' in general.