Ég lokaði glugganum af því að það var mikil rigning.

Breakdown of Ég lokaði glugganum af því að það var mikil rigning.

ég
I
vera
to be
það
it
af því að
because
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Questions & Answers about Ég lokaði glugganum af því að það var mikil rigning.

Why is glugganum in the dative case?
The verb loka (“to close”) in Icelandic is one of those verbs that takes its object in the dative, not the accusative. Moreover, because you’re closing a specific window, it must be in the definite form. For the masculine noun gluggi, the dative singular definite is glugganum.
What’s the difference between af því að, því að and vegna þess að?

All three can mean “because,” but they differ slightly in register and structure:

af því að – the most neutral spoken form (“because”).
því að – a bit more colloquial, you can drop af in casual speech.
vegna þess að – a bit more formal and literal (“due to the fact that”); vegna takes a genitive, hence þess.

You can use any of them to introduce a causal clause, but af því að is very common in everyday speech.

What is the function of það in það var mikil rigning?

Icelandic often uses a dummy or expletive það (“it”) as the grammatical subject in impersonal weather or existential sentences. The real “subject” here is mikil rigning, but Icelandic syntax requires something in the subject slot, so you get:
Það var mikil rigning” = “It was heavy rain.”

Why is the adjective mikil and not mikill?
Adjectives must agree in gender, number and case with the noun, and they have two sets of endings (strong vs. weak). Here you have an indefinite, feminine, nominative noun (rigning), so you use the strong form for feminine nominative singular, which is mikil (not mikill, which is masculine).
Why isn’t there a definite article on rigning (“rigning” rather than “rigningin”)?
You’re talking about “heavy rain” in a general sense, not “the specific rain” you and your listener know about. In Icelandic, indefinite nouns take no article and trigger the strong adjective form. If you wanted to say “the heavy rain,” you’d use hin rigningin or rigningin with a weak adjective.
How is lokaði formed? Is loka a regular verb?

Yes. Loka is a regular weak verb in the “-a” class. Its preterite (simple past) is formed by adding -aði to the stem:
loka- → loka + ði → lokaði (“I closed”).

Could you show me the singular declension of gluggi (both indefinite and definite)?

Indefinite singular
• Nominative: gluggi
• Accusative: glugga
• Dative: glugga
• Genitive: gluggs

Definite singular
• Nominative: glugginn
• Accusative: gluggann
• Dative: glugganum
• Genitive: gluggans

In our sentence, we need the definite dative form, hence glugganum.