Hun læser første kapitel langsomt, fordi hun vil forstå romanen bedre.

Læser is the present tense of at læse (to read).

In Danish, most verbs form the present tense by adding -r to the infinitive:

• at læse = to read
• læser = reads / is reading

Unlike English, Danish verbs do not change according to the subject:

• jeg læser
• du læser
• hun læser
• vi læser

So hun læser simply means she reads / she is reading.

Both can be possible, but første kapitel is very natural when talking about numbered parts such as chapters, pages, sections, rounds, and so on.

With things that are identified by order or number, Danish often allows the article to be left out:

• første kapitel = first chapter
• anden side = second page

If you say det første kapitel, that is also correct, but it sounds a bit more fully definite and explicit: the first chapter.

So in this sentence, første kapitel is a natural, slightly leaner way to say it.

Because Danish usually marks definiteness by adding the definite article to the end of the noun.

So:

• en roman = a novel
• romanen = the novel

This is one of the big differences from English. Instead of putting the before the noun, Danish often adds -en, -et, -ene, or -erne to the noun itself.

Here:

• roman
  • -en = romanen

Because den roman usually does not mean plain the novel by itself.

Normally:

• romanen = the novel

But den roman is usually used more like that novel or for special emphasis/contrast.

Also, when there is an adjective, Danish often uses both a separate article and a definite ending pattern:

• den gode roman = the good novel

So in your sentence, romanen is the normal choice for the novel.

Because langsomt is being used adverbially: it describes how she reads.

• langsom = slow
• langsomt = slowly

In Danish, many adjectives can be used as adverbs by adding -t:

• hurtig → hurtigt
• langsom → langsomt

So:

• en langsom bog = a slow book
• hun læser langsomt = she reads slowly

That is the most neutral word order here.

In Danish, an adverb of manner like langsomt often comes after the object:

• Hun læser første kapitel langsomt.

That sounds very natural: She reads the first chapter slowly.

You may sometimes see other placements, but this version is the standard, unmarked one.

Because vil is a modal verb, and modal verbs in Danish are followed by the bare infinitive.

So:

• hun vil forstå = she wants to understand

Not:

• hun vil at forstå

The same happens with other modal verbs:

• hun kan læse = she can read
• hun skal læse = she must / will read
• hun må læse = she may / must read

So after vil, you use forstå, not at forstå.

Here vil means wants to or intends to, not just simple future.

So:

• hun vil forstå romanen bedre

means that this is her goal or intention: she wants to understand the novel better.

Danish vil can sometimes be translated as will, but in many sentences it is closer to want to.

Because fordi introduces a subordinate clause.

In a main clause, Danish normally uses V2 word order: the finite verb is in second position.

But after fordi, that rule does not apply in the same way. The subordinate clause usually has the more basic order:

subject + finite verb + infinitive/object/etc.

So:

• fordi hun vil forstå romanen bedre

This is normal subordinate-clause order.

A very useful thing to notice is where sentence adverbs go. For example:

• Hun læser første kapitel.
• Hun læser ikke første kapitel.

But:

• fordi hun ikke vil forstå romanen bedre

In subordinate clauses, words like ikke usually come before the finite verb's non-subject part and show the non-V2 pattern clearly.

Bedre means better, and here it works adverbially with forstå:

• forstå romanen bedre = understand the novel better

It is the comparative form of god/godt:

• god = good
• godt = well / good
• bedre = better

Danish uses bedre just like English does in expressions such as:

• synge bedre = sing better
• arbejde bedre = work better
• forstå bedre = understand better

Because it naturally modifies the whole idea of understanding the novel, and Danish often places this kind of adverbial element late in the clause.

So:

• hun vil forstå romanen bedre

sounds very natural.

If you move bedre, the sentence may still be understandable, but the original order is the most normal and idiomatic one.

Yes. Each clause needs its own subject.

The sentence has two clauses:

1. Hun læser første kapitel langsomt
2. fordi hun vil forstå romanen bedre

Even though both clauses refer to the same person, Danish still repeats hun in the second clause. English does the same:

• She reads the first chapter slowly because she wants to understand the novel better.

So the second hun is required.