Hún pakkar regnjakka og regnstígvélum, af því að veðurspáin er slæm.

Pakkar is the present tense, 3rd person singular of pakka (to pack).

So:

• ég pakka = I pack / I am packing
• hún pakkar = she packs / she is packing

The verb does not change for gender, only for person and number.

It can mean either one, depending on context.

Icelandic present tense often covers both:

• she packs
• she is packing

If the wider context is about what she is doing right now, English would often translate it as she is packing.

Because Icelandic nouns change form for case and number.

Here, pakka commonly takes the thing being packed in the dative:

• pakka einhverju = pack something

So instead of the dictionary forms

• regnjakki
• regnstígvél

you get:

• regnjakka = dative singular
• regnstígvélum = dative plural

Because they are different nouns and also different numbers.

Both are in the dative, but:

• regnjakka is singular
• regnstígvélum is plural

Also, different noun types decline differently.
Regnjakki is a masculine noun whose dative singular happens to be regnjakka, while regnstígvél has the dative plural ending -um: regnstígvélum.

Because some Icelandic nouns have the same form in more than one case.

For regnjakki, the forms are:

• nominative singular: regnjakki
• accusative singular: regnjakka
• dative singular: regnjakka

So regnjakka could be accusative or dative in form, but in this sentence it is understood as dative because of the verb pakka.

Icelandic has no indefinite article like English a/an.

So a bare noun can often mean:

• a raincoat
• rain boots
• some rain boots

depending on context.

That is why you see:

• regnjakka
• regnstígvélum

without any separate word meaning a.

Af því að means because.

It introduces the reason:

• Hún pakkar ... af því að ...
• She packs ... because ...

It is best learned as a fixed expression. You may also hear shorter variants such as því að in everyday language.

Because Icelandic usually expresses the by attaching it to the noun as a suffix.

So:

• veðurspá = weather forecast
• veðurspáin = the weather forecast

The ending -in is the definite article here.

Because Icelandic adjectives must agree with the noun they describe in gender, number, and usually case.

Here the noun is:

• veðurspáin = feminine singular

So the adjective must also be feminine singular:

• slæm = bad

Compare:

• slæmur = masculine
• slæm = feminine
• slæmt = neuter

Yes. Icelandic uses compound words very often.

You can break them down like this:

• regn + jakki = rain + jacket/coat
• regn + stígvél = rain + boots
• veður + spá = weather + forecast

Learning to spot these parts is very helpful, because many Icelandic words are built this way.

Because af því að introduces a subordinate clause.

In that clause, the word order is the normal pattern:

• conjunction + subject + verb + complement

So:

• af því að = because
• veðurspáin = the weather forecast
• er = is
• slæm = bad

This gives:

• af því að veðurspáin er slæm

That is the natural order for this kind of clause in Icelandic.