Min nabo siger, at hendes nummerplade blev væk sidste år, så nu tager hun et billede af den hver måned.

Blev væk is a very common Danish expression meaning got lost, went missing, or disappeared.

• blev = past tense of blive = became
• væk = away/gone

So literally it looks like became gone, but in natural English you would usually say:

• her license plate went missing
• her license plate got lost
• her license plate disappeared

It is different from var væk, which means was gone/missing and describes a state rather than the change into that state.

Because Danish sin/sit/sine is only used when the possessor is the subject of the same clause.

In the clause:

at hendes nummerplade blev væk

the subject is nummerplade (the license plate), not hun.

So the owner cannot be referred to with sin here. You need hendes = her.

Compare:

• Hun mistede sin nummerplade. = She lost her own license plate.
  Here hun is the subject, so sin is correct.

• Min nabo siger, at hendes nummerplade blev væk.
  Here the subject inside the at-clause is nummerplade, so hendes is correct.

At introduces a subordinate clause and usually corresponds to English that.

So:

• Min nabo siger, at ... = My neighbor says that ...

In English, that is often omitted. In Danish, at is very common and standard here.

So the structure is:

• main clause: Min nabo siger
• subordinate clause: at hendes nummerplade blev væk sidste år

This is a classic Danish word-order point.

In the at-clause

After at, Danish uses subordinate-clause word order:

• at hendes nummerplade blev væk sidste år

Here the subject hendes nummerplade comes before the verb blev.

In the main clause

Danish main clauses normally follow the verb-second rule. In:

• så nu tager hun et billede af den hver måned

the time word nu comes before the verb, so the finite verb tager must come before the subject hun.

That is why you get:

• nu tager hun not
• nu hun tager

This is very typical Danish main-clause word order.

Nummerplade means license plate, number plate, or registration plate.

It is a compound noun:

• nummer = number
• plade = plate

Danish uses compound nouns very often, so it is normal to see long words like this written as one word.

Because et billede and den refer to two different nouns.

et billede

Billede is a neuter noun, so it takes et:

• et billede = a picture

den

Den refers back to nummerplade, not to billede.

• en nummerplade = a license plate
• therefore the pronoun is den = it

So:

• et billede = a picture
• af den = of it / of the license plate

Because Danish says tage et billede af noget/nogen for take a picture of something/someone.

So:

• tage et billede af den = take a picture of it

This is the normal preposition with billede in this meaning.

You could also say:

• fotografere den = photograph it

But tage et billede af den is very natural everyday Danish.

Hver måned means every month.

• hver = every / each
• måned = month

It works like other common time expressions:

• hver dag = every day
• hver uge = every week
• hver måned = every month

In this sentence it comes near the end, which is a very natural place for a time expression.

Så means so, and nu means now.

Together, så nu means:

• so now
• and so now
• therefore now

It shows a consequence:

• her license plate went missing last year,
• so now she takes a picture of it every month.

So så links the two ideas, and nu emphasizes the present situation.

No. Blev væk only says that it went missing, got lost, or disappeared. It does not by itself say how.

It could mean:

• it was misplaced
• it fell off
• someone took it
• nobody knows what happened

If you wanted to say it was definitely stolen, Danish would more clearly say:

• blev stjålet = was stolen

There are two comma points learners may notice here.

1. Before at

• Min nabo siger, at ...

This comma is often used before a subordinate clause. In modern Danish, you may also see this written without the comma:

• Min nabo siger at ...

Both are seen, depending on comma style.

2. Before så

• ..., så nu tager hun ...

This comma separates the first clause from the next main clause and helps show the pause and connection in meaning.

So the punctuation is normal and natural in written Danish.