Når bøgerne er pakket ud, lægger jeg dem på hylden.

Because Danish usually adds the definite article to the end of the noun.

• bog = book
• bøger = books
• bøgerne = the books

So bøgerne is the definite plural form. Also note that bog has an irregular plural: it becomes bøger, not boger.

This is because the sentence focuses on the books as the thing affected, not on the person doing the unpacking.

• Jeg har pakket bøgerne ud = I have unpacked the books.
• Bøgerne er pakket ud = The books have been unpacked / are unpacked.

So er + past participle here gives a passive/resultative meaning: the unpacking is completed, and now the books are in that state.

Yes. Pakke ud is a common Danish verb + particle combination, similar to English unpack.

Its forms can look different depending on the grammar:

• infinitive: at pakke ud
• present: jeg pakker bøgerne ud
• past participle: pakket ud

So even though it is written as two words, it functions as one verbal idea.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

Here, the sentence starts with the subordinate clause:

Når bøgerne er pakket ud,

After that opening element, the main clause must begin with the verb:

lægger jeg dem på hylden

So the order is not the normal English-style I put. Danish requires inversion after a fronted clause.

Når is used for when, especially in cases like:

• something that happens in the future
• something that happens regularly or whenever a condition is met

So Når bøgerne er pakket ud means something like when/once the books are unpacked.

A useful contrast:

• når = when/whenever, often future or repeated
• da = when, for a specific event in the past

So for a past one-time event, Danish would usually use da, not når.

Because Danish often uses the present tense for future meaning when the context already makes the time clear.

Here, når shows that this happens after another action, so Danish does not need a special future form.

This is very common:

• I morgen går jeg hjemme. = Tomorrow I’m staying home.
• Når han kommer, spiser vi. = When he comes, we’ll eat.

So lægger is grammatically present, but the meaning can easily be future.

Dem is the object form of de.

• de = they
• dem = them

Since the books are the thing being put on the shelf, Danish uses the object form:

lægger jeg dem på hylden

It refers back to bøgerne.

Because a shelf is normally thought of as a surface, so Danish uses på = on.

• på hylden = on the shelf

You would use i if something were inside something. A shelf is usually treated as a flat place you put things on, not a container you put things in.

Just like with bøgerne, Danish usually attaches the definite article to the noun.

• hylde = shelf
• hylden = the shelf

The ending -en is the definite article for many common-gender singular nouns.

So på hylden means on the shelf.

Because the sentence begins with a subordinate clause:

Når bøgerne er pakket ud,

and then moves to the main clause:

lægger jeg dem på hylden.

In Danish, commas are commonly used to mark the boundary between these parts. So the comma helps show that the first part is the time clause and the second part is the main statement.

Lægge means to lay / put / place, and it is a normal verb for putting something somewhere.

Danish often makes distinctions like these:

• lægge = lay something down
• stille = put something standing
• sætte = set something down / place it

With books, context matters. If you imagine stacking them flat, lægge fits very naturally. If you imagine placing them upright on a shelf, some speakers might also use stille. So the sentence is perfectly understandable, but Danish sometimes chooses different placement verbs depending on the position of the object.