Ik leg het boek weer in de kast.

Dutch often uses different verbs for put depending on the position of the object after you place it.

• leggen = to lay something down, so the object ends up lying
• zetten = to set/place something, so the object ends up standing
• stoppen = to put something into something in a more general sense, often like stuff/put

With een boek, Dutch often uses leggen, because a book is normally thought of as ending up lying on a surface or inside something.

So:

• Ik leg het boek in de kast. = I put the book in the cupboard/bookcase.
• Ik zet de fles op tafel. = I put the bottle on the table.

This is one of those very Dutch patterns where the choice of verb depends on the object's final position.

Because the book is being placed inside the cupboard/bookcase, not just moved toward it.

• in de kast = in / into the cupboard
• naar de kast = toward the cupboard

In Dutch, with verbs like leggen, zetten, and stoppen, the preposition often shows the final location directly:

• Ik leg het boek in de kast. = I put the book into the cupboard.
• Ik zet de bloemen op tafel. = I put the flowers on the table.

Even though English often uses into, Dutch commonly just uses in here.

It can effectively mean into in this kind of sentence.

That can feel strange to English speakers, because English usually distinguishes:

• The book is in the cupboard. = location
• I put the book into the cupboard. = movement

Dutch often uses the same preposition in for both, while the verb shows whether it is a state or a movement:

• Het boek ligt in de kast. = The book is in the cupboard.
• Ik leg het boek in de kast. = I put the book into the cupboard.

So in your sentence, in de kast is best understood as into the cupboard/bookcase.

Weer usually means again, but in many contexts it can also feel like back.

So:

• Ik leg het boek weer in de kast.

can mean:

• I’m putting the book in the cupboard again
• or I’m putting the book back in the cupboard

The exact nuance depends on context.

If you want to make the idea of back clearer, Dutch often uses terug:

• Ik leg het boek terug in de kast. = I put the book back in the cupboard.

So weer is possible, but terug is often more explicit if back is the important idea.

This is normal Dutch word order in a main clause.

The sentence is structured like this:

• Ik = subject
• leg = finite verb
• het boek = direct object
• weer = adverb
• in de kast = prepositional phrase

Dutch main clauses usually put the finite verb in second position:

• Ik leg ...

After that, the rest often goes in the middle field, where adverbs like weer commonly come before longer prepositional phrases such as in de kast.

So Ik leg het boek weer in de kast sounds very natural.

You may also hear variation depending on emphasis, but this version is the most neutral.

It is not the most natural neutral order.

Dutch usually prefers het boek weer rather than weer het boek in a sentence like this:

• Ik leg het boek weer in de kast. = natural
• Ik leg weer het boek in de kast. = possible only in special emphasis contexts, but not the normal choice

Putting weer before het boek can sound marked, as if you are contrasting it with something else.

So for everyday Dutch, stick with:

• Ik leg het boek weer in de kast.

Because Dutch nouns have two grammatical genders in modern standard Dutch:

• de-words
• het-words

In this sentence:

• boek is a het-word → het boek
• kast is a de-word → de kast

Unfortunately, this usually has to be learned with each noun.

A useful tip:

• most singular nouns take de
• all diminutives take het
• but there are many common non-diminutive het-words, like het boek, het huis, het kind

So this sentence gives you one of each:

• het boek
• de kast

Kast is a broad Dutch word. Depending on context, it can mean:

• cupboard
• cabinet
• closet
• wardrobe
• bookcase

So de kast does not always mean exactly the same thing in English.

With boek, many learners imagine bookcase or cabinet, but the exact translation depends on the situation.

That is very normal in Dutch: one word can cover several English words.

Yes, leggen normally takes a direct object: you lay/put something somewhere.

In this sentence, the thing being placed is:

• het boek

And the destination is:

• in de kast

So the pattern is:

• iemand legt iets ergens
  someone puts something somewhere

Examples:

• Ik leg de sleutels op tafel.
• Zij legt de krant op de stoel.

By contrast, the matching intransitive verb is usually liggen:

• Het boek ligt in de kast. = The book is lying in the cupboard.

This is an important Dutch pair:

• leggen = to lay/put
• liggen = to lie/be lying

This is a very common Dutch difficulty.

• leggen = to lay / put something down
  → transitive, so it takes an object
• liggen = to lie / be lying
  → intransitive, describing the position

Compare:

• Ik leg het boek in de kast. = I put the book in the cupboard.
• Het boek ligt in de kast. = The book is in the cupboard / lying in the cupboard.

So leggen is the action of placing it, and liggen is the resulting position.

Yes. Dutch simple present tense often covers several English possibilities, depending on context.

Ik leg het boek weer in de kast can mean things like:

• I put the book back in the cupboard.
• I am putting the book back in the cupboard.
• I put the book in the cupboard again.

Dutch does not always need a special progressive form like English am putting.

If context shows it is happening right now, the simple present is enough.

Yes, and in many situations terug may even be more precise.

Compare:

• weer = again, and sometimes back
• terug = back, back again, returning to an earlier place/state

So:

• Ik leg het boek weer in de kast. = I put the book in the cupboard again / back
• Ik leg het boek terug in de kast. = I put the book back in the cupboard

If the idea is specifically that the book was out of the cupboard and is now being returned, terug is often the clearest choice.

Yes. Weer can also mean weather, but that is a completely different word in function, even though it is spelled the same.

Examples:

• Het weer is mooi. = The weather is nice.
• Ik doe het weer. = I do it again.

In your sentence, weer is the adverb meaning again/back, not the noun meaning weather.

Because Dutch does not need one here.

English often says:

• I put the book into the cupboard

Dutch expresses this with the placement verb plus the prepositional phrase:

• Ik leg het boek in de kast

So the idea of movement toward a final place is already clear from:

• the verb leg
• the phrase in de kast

Dutch simply packages this differently from English.