Metto il libro nello zaino.

In Italian, the subject pronoun is often left out because the verb ending already shows who is doing the action.

• metto = I put / I am putting
• The -o ending tells you it is I

So Metto il libro nello zaino is completely natural.
You could say Io metto il libro nello zaino, but that usually adds emphasis, as in I am putting the book in the backpack.

Metto is the present tense, first person singular, of mettere.

It can mean different things in English depending on context, such as:

• I put
• I am putting
• sometimes even I do put

Italian present tense often covers both the simple present and the present continuous that English keeps separate.

Mettere is a very common verb that usually means:

• to put
• to place
• to set

In this sentence, it means to put something somewhere.

Some common forms are:

• metto = I put
• metti = you put
• mette = he/she puts
• mettiamo = we put
• mettono = they put

It is a very useful verb and appears in many expressions.

Il is the masculine singular definite article, meaning the.

So:

• il libro = the book

Italian uses articles very regularly, often more often than English does. In this sentence, the speaker is talking about a specific book, so il libro is the normal choice.

Because nello is a combination of:

• in
  • lo = nello

Italian often combines certain prepositions with definite articles.

Here, zaino takes the article lo, so:

• in lo zaino becomes nello zaino

This is not optional in standard Italian; the combined form is the correct one.

Some masculine singular nouns take lo instead of il.

Lo is used before words beginning with:

• z
• s + consonant
• ps
• gn
• x
• sometimes y

Since zaino begins with z, it takes lo:

• lo zaino

That is why:

• in + lo zaino = nello zaino

In this sentence, it most naturally means into the backpack, because there is movement from one place to another.

With verbs like mettere, Italian often uses in + article to show movement into something:

• Metto il libro nello zaino = I put the book into the backpack

In English, we may also simply say in the backpack, but the idea is still movement inward.

Yes, it is somewhat irregular, and it is worth learning well because it is very common.

A few important points:

• infinitive: mettere
• past participle: messo
• present: metto, metti, mette, mettiamo, mettete, mettono

So the stem changes a little in some forms, and the past participle is not predictable from the infinitive.

Yes, Italian word order is flexible, although this sentence uses a very normal and neutral order.

Standard neutral version:

• Metto il libro nello zaino.

Possible variations:

• Il libro lo metto nello zaino.
  This emphasizes the book.
• Nello zaino metto il libro.
  This emphasizes where you put it.

So the original sentence is the most straightforward everyday version.

Zaino is pronounced roughly like DZAI-no.

A few helpful notes:

• z here sounds like dz
• ai sounds like the vowel sound in eye
• stress falls on the first syllable: ZAI-no

So: DZAI-no

Yes, you can, but it is a little different in structure.

• nello zaino = in/into the backpack
• dentro lo zaino = inside the backpack

With mettere, both can work, but nello zaino is the more direct and natural choice in a simple sentence like this.

For example:

• Metto il libro nello zaino.
• Metto il libro dentro lo zaino.

Both are understandable, but the first one is especially common.

Italian normally uses the article with nouns in cases where English may or may not use it.

Here Italian says:

• nello zaino = literally in the backpack

That article is part of the normal grammar. Since the noun is specific and countable, Italian usually includes it. So even if English phrasing can vary a little, in Italian the article is expected here.