Hon skriver i sin dagbok varje kväll innan hon lägger sig.

Because sin is the reflexive possessive pronoun. Swedish uses sin / sitt / sina when the possessor is the same as the subject of the clause.

In Hon skriver i sin dagbok, the subject is hon, so sin dagbok means her own diary.

If you said hennes dagbok, it would usually suggest someone else’s diary, not the subject’s own diary.

Also, dagbok is an en-word, so the correct form is sin. The full set is:

sin for common gender singular
sitt for neuter singular
sina for plural

In Swedish, you normally do not use an article together with a possessive.

So Swedish says:

sin dagbok = her diary
not sin en dagbok

This is the same pattern as:

min bok = my book
hans bil = his car
vårt hus = our house

The possessive already makes the noun definite enough, so no separate article is needed.

Because Swedish idiomatically says skriva i en dagbok = write in a diary.

So:

Hon skriver i sin dagbok = She writes in her diary

This works much like English here, but it is still worth learning as a fixed pattern: skriva i when you mean writing inside a notebook, diary, journal, or similar written space.

Skriver is the present tense of skriva.

Here it expresses a habitual action, so it means something like writes rather than is writing. The phrase varje kväll makes that especially clear.

So the sentence means that this is something she does regularly, not just right now.

Swedish present tense is often used for: general truths, habits, and sometimes actions happening now, depending on context.

Yes, it could move, but the placement here is very natural.

The sentence has this basic flow:

Hon skriver + i sin dagbok + varje kväll + innan hon lägger sig

That is: subject + verb + place/context + time + subordinate clause

You could also say:

Varje kväll skriver hon i sin dagbok innan hon lägger sig.

That version puts emphasis on every evening. Because Swedish main clauses follow the verb-second rule, when Varje kväll moves to the front, the verb skriver must come immediately after it.

Innan means before.

It introduces a subordinate clause:

innan hon lägger sig = before she goes to bed / before she lies down

So the whole sentence says that the writing happens before the bedtime action.

Because innan hon lägger sig is a full subordinate clause, and that clause needs its own subject.

So Swedish, like English, repeats the subject:

before she goes to bed

not just before goes to bed

Even though it is the same person as earlier in the sentence, the new clause still needs hon.

Because after innan, Swedish uses a subordinate clause, and subordinate clauses do not follow the same verb-second pattern as main clauses.

So the normal order is:

innan + subject + verb

That gives:

innan hon lägger sig

This is different from a main clause, where Swedish often places the verb in second position.

So this sentence is a good example of the contrast between: main clause word order and subordinate clause word order.

Lägger sig is the present tense of lägga sig, a reflexive verb meaning lie down or, very often in everyday Swedish, go to bed.

The word sig is a reflexive pronoun, referring back to the subject hon.

So:

hon lägger sig = she lies down / she goes to bed

Without sig, lägger usually means puts or places something:

Hon lägger boken på bordet = She puts the book on the table

So lägger sig and lägger are not the same thing.

This is a very common confusion.

Swedish distinguishes between: lägga sig = to lie down, to go to bed, the action of changing position
ligga = to be lying, the state after the movement

So in this sentence, she is not already in bed. The meaning is that she writes before the action of going to bed. That is why lägger sig is correct.

Compare:

Hon lägger sig klockan tio. = She goes to bed at ten.
Hon ligger i sängen. = She is lying in bed.

Yes. That is also very common and natural.

innan hon lägger sig is a bit shorter and more direct.
innan hon går och lägger sig is slightly more explicit, like before she goes and lies down / before she goes to bed.

Both are idiomatic, but the version in your sentence is completely normal and natural Swedish.