Hun siger, at huslejen er for høj, så hun bliver i sin gamle lejlighed.

At means that here.

So Hun siger, at ... = She says that ...

In Danish, at often introduces a subordinate clause after verbs like:

• sige = say
• tro = think/believe
• vide = know
• mene = mean/think

In everyday speech, Danish speakers sometimes leave at out, just like English sometimes drops that:

• Hun siger, huslejen er for høj.

But in writing, keeping at is very common and often clearer.

Husleje means rent, while huslejen means the rent.

The ending -en is the definite article added to the noun:

• en husleje = a rent / rent
• huslejen = the rent

Danish usually attaches the to the end of the noun instead of putting a separate word in front.

So:

• bilen = the car
• bogen = the book
• lejligheden = the apartment
• huslejen = the rent

Danish often uses høj = high for prices, costs, taxes, rent, etc., just like English does in phrases such as the rent is too high.

So:

• Prisen er høj = The price is high
• Skatten er høj = The tax is high
• Huslejen er for høj = The rent is too high

Here for means too:

• for dyr = too expensive
• for høj = too high

Both can be possible in some contexts, but with rent, price, salary, temperature, and similar things, høj is very natural.

Here så means so in the sense of therefore / as a result.

So the structure is:

• huslejen er for høj = the rent is too high
• så hun bliver i sin gamle lejlighed = so she stays in her old apartment

It connects cause and result.

Be careful: så can also mean other things in Danish depending on context, such as:

• then
• so
• part of set phrases

But in this sentence, it clearly means so / therefore.

Yes, blive often means become, but it can also mean stay / remain in certain contexts.

Here:

• hun bliver i sin gamle lejlighed

means

• she stays in her old apartment

So blive has more than one common use:

• Han bliver træt. = He becomes tired.
• Jeg bliver hjemme. = I stay home.
• Hun bliver i lejligheden. = She stays in the apartment.

Bliver is the present tense form of blive.

Because Danish uses a reflexive possessive when the owner is the subject of the same clause.

In the clause:

• så hun bliver i sin gamle lejlighed

the subject is hun, and the apartment belongs to her, so Danish uses sin.

This is a very important difference from English.

Reflexive possessive:

• sin / sit / sine = his/her/its/their own

Non-reflexive possessive:

• hendes = her
• hans = his
• deres = their

Compare:

• Hun bliver i sin gamle lejlighed.
  = She stays in her own old apartment.

• Hun bliver i hendes gamle lejlighed.
  = She stays in another woman’s old apartment, or at least not necessarily her own.

So sin shows that the apartment belongs to the subject hun.

Because the form depends on the gender and number of the noun being possessed.

The noun is:

• lejlighed = apartment

And lejlighed is a common gender singular noun (en lejlighed), so the correct reflexive possessive is sin.

The pattern is:

• sin
  • common gender singular noun
• sit
  • neuter singular noun
• sine
  • plural noun

Examples:

• sin lejlighed = her/his own apartment
• sit hus = her/his own house
• sine bøger = her/his own books

Because adjectives usually take the -e form after possessives like sin, min, din, hans, hendes, etc.

So:

• en gammel lejlighed = an old apartment
• sin gamle lejlighed = her own old apartment

After a possessive, Danish typically uses the definite/weak adjective form, which is often the -e form.

Other examples:

• min gamle bil = my old car
• hendes nye job = her new job
• deres store hus = their big house

So gamle is the expected form here.

It does, and this is a good example of that.

After at, you have a subordinate clause, and subordinate clauses do not follow the normal Danish main-clause verb-second rule.

So:

Main clause:

• Huslejen er for høj.

Subordinate clause:

• ... at huslejen er for høj

The subject comes before the verb:
huslejen er

That is normal subordinate-clause order.

Because så here is acting as a conjunction meaning so, connecting two clauses.

After a conjunction like this, the next clause can begin normally with the subject:

• så hun bliver i sin gamle lejlighed

This is different from sentence adverbs or fronted elements that trigger inversion in Danish main clauses.

A learner often notices that Danish sometimes does this:

• Nu bliver hun hjemme.
• Derfor bliver hun hjemme.

Here the verb comes before the subject because Nu and Derfor are fronted adverbials in a main clause.

But in your sentence, så works more like so, linking the two clauses, so hun bliver is natural.

I means in here.

So:

• bliver i lejligheden = stays in the apartment

Danish uses i with enclosed spaces, buildings, rooms, cities, countries, etc., much like English uses in.

Examples:

• i huset = in the house
• i skolen = in the school
• i København = in Copenhagen

So hun bliver i sin gamle lejlighed literally means she stays in her old apartment.

Because the clause starting with at is a subordinate clause, and Danish spelling traditionally uses a comma before such clauses.

So:

• Hun siger, at huslejen er for høj ...

This comma helps show where the subordinate clause begins.

You may also notice the comma before så in this sentence is not used. That is normal here, because the sentence is written as one flowing structure with coordination by så.

Comma rules in Danish can vary a little depending on the comma system being used, but the comma before at is very common and standard.

A natural rhythm would usually stress the important content words:

Hun siger, at HUSlejen er for HØJ, så hun BLIVER i sin GAMle lejlighed.

Likely pronunciation notes:

• hun sounds roughly like hoon, but shorter and more centralized
• siger is roughly SEE-er / SI-er
• huslejen is roughly HOOS-lie-en in broad learner-friendly terms
• høj has a rounded vowel, something like hoy but not exactly English
• bliver is roughly BLEE-ver or BLI-ver depending on speed
• lejlighed is challenging: roughly LIE-lig-hade, but the real pronunciation is more reduced
• gamle sounds roughly gam-luh

In connected speech, function words like at, så, i, and sin are usually less stressed than the main nouns, adjectives, and verbs.