Heimaþjónustan byrjar á morgun.

Because Icelandic very often makes compound nouns by joining words together.

• heima- = home- / at home
• þjónusta = service

So heimaþjónusta is literally something like home-service. This is completely normal in Icelandic. English sometimes writes similar ideas as two words, with a hyphen, or as one word, but Icelandic usually prefers a single compound word.

It marks the noun as definite, so it means the home service, not just home service.

A key Icelandic difference from English is that the definite article is usually attached to the end of the noun instead of being a separate word:

• heimaþjónusta = home service
• heimaþjónustan = the home service

With feminine nouns ending in -a, this often shows up as -n attached at the end:

• þjónusta → þjónustan

It is feminine.

In Icelandic compounds, the last part usually decides the gender. Here the last part is þjónusta, and þjónusta is a feminine noun. So the whole compound heimaþjónusta is feminine too.

That is also why the definite form is heimaþjónustan, following a common feminine pattern.

Because byrjar is the correct present-tense finite form for a third-person singular subject.

• að byrja = to begin / to start
• heimaþjónustan = the home service → grammatically it

So Icelandic uses:

• Heimaþjónustan byrjar = The home service starts

The form byrja is the infinitive (to start) and is also used in some other present-tense forms, but not with this singular subject here.

Because á morgun is a fixed Icelandic expression meaning tomorrow.

Even though morgun is related to morning, the whole phrase á morgun does not just mean in the morning. It means tomorrow as a time reference.

You can think of it as an idiom that simply has to be learned as a set phrase.

This is a very common point of confusion:

• á morgun = tomorrow
• í morgun = this morning

So:

• Heimaþjónustan byrjar á morgun. = The home service starts tomorrow.
• Heimaþjónustan byrjaði í morgun. = The home service started this morning.

The preposition makes a big difference here.

Here it is best to learn á morgun as a fixed expression, but grammatically á can govern different cases in different uses, and in this expression morgun is traditionally analyzed as accusative.

That matters because Icelandic prepositions often affect case. However, for a learner, the most important thing is simply:

• memorize á morgun = tomorrow

rather than trying to build it from scratch every time.

Yes. Icelandic word order is flexible, but the finite verb usually stays in second position in main clauses.

The given sentence is a neutral order:

• Heimaþjónustan byrjar á morgun.

You can also front the time expression:

• Á morgun byrjar heimaþjónustan.

That is also correct, and it may put a little more emphasis on tomorrow.

What you normally do not do in a standard main clause is move the time phrase first and then leave the verb later, as English sometimes seems to allow. Icelandic strongly prefers that byrjar stay in the second slot.

A rough English-friendly guide is:

• Heimaþjónustan ≈ HAY-ma-thyOH-nu-stan
• byrjar ≈ BIR-yar
• á morgun ≈ ow MOR-gun

A few useful pronunciation points:

• þ sounds like th in thin
• j sounds like y in yes
• á sounds roughly like ow in cow
• Icelandic stress usually falls on the first syllable

So the whole sentence is roughly:

HAY-ma-thyOH-nu-stan BIR-yar ow MOR-gun

That is only an approximation, but it is a good starting point for an English speaker.