Musikken er så lav, at jeg næsten ikke kan høre den.

Musikken is the definite form, meaning the music.

• musik = music
• musikken = the music

In Danish, the definite article is usually attached to the end of the noun instead of being a separate word:

• en bog = a book
• bogen = the book

So Musikken er så lav ... means The music is so quiet/low ...

Yes. Lav literally means low, but in Danish it is also commonly used about volume.

So:

• lav musik = low/quiet music
• høj musik = loud music

English usually says quiet or soft, while Danish often says low in this context.

Så ... at ... means so ... that ...

In this sentence:

• så lav = so low / so quiet
• at jeg næsten ikke kan høre den = that I can hardly hear it

This is a very common Danish structure:

• Han er så træt, at han går i seng nu.
  = He is so tired that he is going to bed now.
• Det er så koldt, at jeg bliver inde.
  = It is so cold that I’m staying inside.

Because Danish spelling normally uses a comma before subordinate clauses, and at here introduces one.

So the sentence is divided like this:

• main clause: Musikken er så lav
• subordinate clause: at jeg næsten ikke kan høre den

This comma is standard in written Danish.

This is because Danish has a typical word order pattern in subordinate clauses.

In subordinate clauses, the sentence adverbial such as ikke, næsten, aldrig, ofte, etc. usually comes before the finite verb.

So:

• at jeg næsten ikke kan høre den

Here:

• jeg = subject
• næsten ikke = almost not / hardly
• kan = finite verb
• høre = infinitive
• den = object pronoun

Compare:

• main clause: Jeg kan næsten ikke høre den.
• subordinate clause: ... at jeg næsten ikke kan høre den.

This difference is very important in Danish grammar.

Næsten ikke literally means almost not, but in natural English it usually becomes:

• hardly
• almost not
• can barely

So:

• jeg kan næsten ikke høre den
  = I can hardly hear it
  = I can barely hear it

It is a very common Danish expression.

Examples:

• Jeg sover næsten ikke. = I hardly sleep.
• Han spiser næsten ikke noget. = He hardly eats anything.

Kan høre means can hear / am able to hear, which emphasizes ability.

• jeg hører den = I hear it
• jeg kan høre den = I can hear it

In this sentence, the point is that the music is so quiet that hearing it is difficult, so kan høre is the natural choice.

Den means it, and it refers back to musikken.

Since musik is a common-gender noun (en musik is not used as a count noun, but grammatically the word takes common-gender agreement in definite reference), the pronoun is den rather than det.

So:

• musikken = the music
• den = it

Danish often repeats the noun as a pronoun when it becomes the object later in the sentence:

• Jeg hører musikken. = I hear the music.
• Jeg hører den. = I hear it.

Because den is the object of høre.

In Danish, with a modal verb plus infinitive, the structure is typically:

• subject + modal verb + infinitive + object

So:

• jeg kan høre den
• jeg vil købe den
• hun må tage den

That is why den comes after høre here.

Yes, lav is correct because it agrees with musikken, which is a common-gender singular noun.

Adjective agreement in Danish works like this:

• common gender singular: lav
• neuter singular: lavt
• plural / definite: lave

Examples:

• musikken er lav = the music is low/quiet
• lydniveauet er lavt = the sound level is low
• de lave toner = the low tones

So lavt would not fit with musikken.

Sometimes, but not in exactly the same way.

• lav is the normal word for low/quiet volume
• stille usually means quiet in the sense of silent/calm/not making much noise

So for volume, Danish usually prefers lav:

• Musikken er lav. = The music is quiet/low.

If you say Musikken er stille, it sounds less natural in this context.

The structure is:

• Musikken = subject
• er = verb
• så lav = complement
• at jeg næsten ikke kan høre den = subordinate result clause

So the sentence is built like this:

[Subject] + [verb] + så + adjective + at + [subordinate clause]

A useful model to remember is:

• Det er så dyrt, at jeg ikke kan købe det.
  = It is so expensive that I can’t buy it.
• Hun taler så hurtigt, at jeg ikke forstår hende.
  = She speaks so fast that I don’t understand her.

In natural spoken Danish, it would often be pronounced quite smoothly, with some words reduced.

A learner mainly needs to notice these points:

• Musikken often sounds more like musiggen in fast speech.
• er is usually very weak.
• at in the middle may be pronounced lightly, sometimes almost like just a short vowel.
• kan may also be reduced in casual speech.

But the written form remains:

Musikken er så lav, at jeg næsten ikke kan høre den.

So even if spoken Danish sounds compressed, the grammar and spelling here are completely standard.