Eftir langa vakt vill hún bara fara heim og hvíla sig.

Because the preposition eftir takes the accusative when it means after in a time sense.

• vakt is a feminine noun.
• The accusative singular form of löng is langa.
• So:
  • nominative: löng vakt
  • accusative: langa vakt

That is why Icelandic says eftir langa vakt = after a long shift.

Here eftir means after.

In this sentence it refers to time:

• Eftir langa vakt = After a long shift

This is a very common use of eftir with the accusative:

• eftir mat = after dinner
• eftir vinnu = after work
• eftir langan dag = after a long day

Vakt usually means shift, especially a work shift.

Common examples:

• á vakt = on duty / on shift
• næturvakt = night shift
• dagvakt = day shift

So langa vakt suggests she has been working for a long time and is tired.

The verb is vilja = to want.

Its present-tense forms are irregular:

• ég vil = I want
• þú vilt = you want
• hann / hún / það vill = he / she / it wants
• við viljum = we want
• þið viljið = you (plural) want
• þau vilja = they want

So hún vill is simply the correct 3rd person singular form: she wants.

Because after vilja + another verb, Icelandic normally uses the bare infinitive without að.

So:

• hún vill fara = she wants to go
• ég vil borða = I want to eat
• þau vilja sofa = they want to sleep

This is similar to English after want: she wants to go, not she wants to to go.

Bara usually means just, only, or sometimes simply.

Here it adds the sense:

• She just wants to go home and rest.

It emphasizes that this is all she wants after the long shift.

Compare:

• Hún vill fara heim. = She wants to go home.
• Hún vill bara fara heim. = She just wants to go home.

Because heim is an adverb meaning home(wards), not a noun with a preposition here.

So:

• fara heim = go home
• koma heim = come home

This works a lot like English, where we also usually say go home, not go to home.

Related forms:

• heima = at home
• heim = home, toward home

So:

• Hún er heima. = She is at home.
• Hún fer heim. = She goes home.

Sig is a reflexive pronoun, meaning that the action goes back to the subject.

So hvíla sig means:

• rest oneself
• more naturally in English: rest

In Icelandic, many verbs are commonly used with a reflexive pronoun where English would not use one.

Here:

• hún ... hvíla sig = she ... rest herself / rest

Because the subject is hún, the reflexive pronoun is sig.

Yes. This is a chain of two infinitives joined by og (and):

• fara heim = go home
• hvíla sig = rest

Together:

• bara fara heim og hvíla sig = just go home and rest

Both infinitives depend on vill:

• hún vill fara heim
• hún vill hvíla sig

Instead of repeating vill, Icelandic links the two infinitives with og.

The sentence begins with a prepositional phrase:

• Eftir langa vakt = After a long shift

Then comes the main clause:

• vill hún bara fara heim og hvíla sig

Notice that the verb vill comes before the subject hún. That happens because Icelandic is a V2 language: in main clauses, the finite verb usually comes in the second position.

So the structure is roughly:

1. Eftir langa vakt
2. vill
3. hún
4. the rest

If you started with the subject instead, you would get:

• Hún vill bara fara heim og hvíla sig eftir langa vakt.

That is also understandable, but the original sentence emphasizes the time phrase first: After a long shift...

It can appear both ways, but hvíla sig is very common when talking about a person resting.

• hvíla sig = rest, take a rest
• hvíla can also mean rest something or let something lie/rest

Examples:

• Ég þarf að hvíla mig. = I need to rest.
• Hún vill hvíla sig. = She wants to rest.
• Við hvílum fæturna. = We rest our feet.

So in this sentence, hvíla sig is the natural choice because the subject herself is the one resting.

A natural rhythm would put stress on langa vakt, bara, heim, and hvíla sig, depending on what the speaker wants to emphasize.

A learner should especially notice these pronunciation points:

• langa: the g is not pronounced like hard English g in all contexts; Icelandic consonants can sound softer than expected.
• vill: sounds roughly like vitl to English ears, though not exactly.
• hún: the accented ú is a long vowel.
• heim: pronounced somewhat like haym.
• hvíla: the hv combination has a distinctive Icelandic sound, often close to kv for many speakers.

But grammatically, the most important thing is recognizing the chunks:

• Eftir langa vakt
• vill hún bara
• fara heim og hvíla sig