Det var först hemma som hon märkte att hennes utkast inte hade sparats.

This is a very common Swedish cleft sentence pattern:

Det var X som Y

It is used to highlight or emphasize one part of the sentence. Here, the highlighted part is först hemma.

So:

Det var först hemma som hon märkte ...

means something like:

It was only when she was home that she noticed ...

The det here is a dummy subject. It does not refer to a specific thing. It is just part of the structure.

A more neutral version would be:

Hon märkte först hemma att hennes utkast inte hade sparats.

The cleft version puts extra focus on först hemma.

In this sentence, först does not simply mean first in a sequence. It has the sense of only then or not until.

So:

först hemma
means roughly
only once she was home / not until she got home

This is a very common Swedish use of först.

Compare:

• Jag förstod det först senare. = I didn’t understand it until later.
• Hon märkte det först hemma. = She didn’t notice it until she was home.

So the sentence is emphasizing the moment when she became aware of the problem.

In this cleft construction, som introduces the rest of the clause after the focused element.

Pattern:

Det var X som ...

So in:

Det var först hemma som hon märkte ...

som is a normal and expected part of the structure. It is not functioning exactly like English who or which here. It is more of a grammatical link in the cleft sentence.

If you remove som, the sentence becomes ungrammatical.

Because hemma is an adverb, not a noun phrase.

It means at home or home, depending on context.

So Swedish says:

• Jag är hemma. = I am at home.
• Hon kom hem. = She came home.
• Han stannade hemma. = He stayed at home.

You do not use a preposition with hemma.

A useful contrast is:

• hem = home, toward home
• hemma = at home

So here först hemma means only once she was at home.

Märkte is the past tense of märka.

märka can mean notice, realize, or become aware of, depending on context.

Here, märkte is used because the sentence is describing a past event in a narrative:

• first she got home
• then she noticed the draft had not been saved

So:

hon märkte = she noticed / she realized

A closely related verb is inse = realize, but märka often suggests noticing something, sometimes suddenly or through evidence.

This is a very important Swedish grammar point.

Swedish uses sin / sitt / sina only when the possessor refers back to the subject of the same clause.

Here the relevant clause is:

att hennes utkast inte hade sparats

In that clause, the grammatical subject is:

hennes utkast

not hon

So hon is not the subject of this subordinate clause, and therefore you cannot use sitt to refer back to her here.

That is why Swedish uses:

hennes utkast = her draft

not

sitt utkast

Even though in meaning the draft belongs to her, the clause structure matters.

Because Swedish possessives usually replace the article.

So you say:

• hennes utkast = her draft
• min bok = my book
• deras bil = their car

not:

• det hennes utkast
• ett hennes utkast

This works much like English: you say her draft, not the her draft.

Also, utkast is the noun here, and hennes already tells us whose draft it is.

Because this is a subordinate clause, introduced by att.

In Swedish subordinate clauses, sentence adverbs like inte usually come before the finite verb.

So:

• Main clause: Utkastet hade inte sparats.
• Subordinate clause: ... att utkastet inte hade sparats.

That is a standard word order difference in Swedish.

A simple way to remember it:

• main clause: verb before inte
• subordinate clause: inte before the finite verb

Hade sparats is a past perfect passive form.

It corresponds to English:

had been saved

Breakdown:

• hade = had
• sparats = been saved / saved, in a passive form

So:

inte hade sparats
means
had not been saved

The sentence is talking about something that had already failed to happen before she noticed it. That is why the past perfect is used.

The -s marks a passive form.

The base verb is:

spara = to save

Passive forms can be made in Swedish with -s:

• sparas = be saved / is saved
• sparades = was saved
• sparats = been saved / had been saved, depending on context with an auxiliary

So inte hade sparats literally contains that passive idea: the draft had not been saved.

Swedish often uses this -s passive where English uses be + past participle.

Yes. A more neutral version would be:

Hon märkte först hemma att hennes utkast inte hade sparats.

This means essentially the same thing.

The difference is emphasis:

• Det var först hemma som hon märkte ...
  strongly highlights först hemma
• Hon märkte först hemma att ...
  is more straightforward and less emphatic

So the original sentence is chosen because it puts special focus on the fact that she only noticed once she was home.